Cálculo Exemplos

$y x^{4} d x = (1 + x^{2}) y^{\frac{1}{2}} d y$

Etapa 1

Reescreva a equação.

$(1 + x^{2}) y^{\frac{1}{2}} d y = y x^{4} d x$

Etapa 2

Multiplique os dois lados por $\frac{1}{(1 + x^{2}) y}$ .

$\frac{1}{(1 + x^{2}) y} ((1 + x^{2}) y^{\frac{1}{2}}) d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3

Simplifique.

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Etapa 3.1

Cancele o fator comum de $1 + x^{2}$ .

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Etapa 3.1.1

Fatore $1 + x^{2}$ de $(1 + x^{2}) y$ .

$\frac{1}{(1 + x^{2}) (y)} ((1 + x^{2}) y^{\frac{1}{2}}) d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.1.2

Fatore $1 + x^{2}$ de $(1 + x^{2}) y^{\frac{1}{2}}$ .

$\frac{1}{(1 + x^{2}) (y)} ((1 + x^{2}) (y^{\frac{1}{2}})) d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.1.3

Cancele o fator comum.

$\frac{1}{(1 + x^{2}) y} ((1 + x^{2}) y^{\frac{1}{2}}) d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.1.4

Reescreva a expressão.

$\frac{1}{y} y^{\frac{1}{2}} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.2

Combine $\frac{1}{y}$ e $y^{\frac{1}{2}}$ .

$\frac{y^{\frac{1}{2}}}{y} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.3

Mova $y^{\frac{1}{2}}$ para o denominador usando a regra do expoente negativo $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{y \cdot y^{- \frac{1}{2}}} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.4

Multiplique $y$ por $y^{- \frac{1}{2}}$ somando os expoentes.

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Etapa 3.4.1

Multiplique $y$ por $y^{- \frac{1}{2}}$ .

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Etapa 3.4.1.1

Eleve $y$ à potência de $1$ .

$\frac{1}{y^{1} y^{- \frac{1}{2}}} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.4.1.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{1}{y^{1 - \frac{1}{2}}} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.4.2

Escreva $1$ como uma fração com um denominador comum.

$\frac{1}{y^{\frac{2}{2} - \frac{1}{2}}} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.4.3

Combine os numeradores em relação ao denominador comum.

$\frac{1}{y^{\frac{2 - 1}{2}}} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.4.4

Subtraia $1$ de $2$ .

$\frac{1}{y^{\frac{1}{2}}} d y = \frac{1}{(1 + x^{2}) y} (y x^{4}) d x$

Etapa 3.5

Cancele o fator comum de $y$ .

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Etapa 3.5.1

Fatore $y$ de $(1 + x^{2}) y$ .

$\frac{1}{y^{\frac{1}{2}}} d y = \frac{1}{y (1 + x^{2})} (y x^{4}) d x$

Etapa 3.5.2

Fatore $y$ de $y x^{4}$ .

$\frac{1}{y^{\frac{1}{2}}} d y = \frac{1}{y (1 + x^{2})} (y (x^{4})) d x$

Etapa 3.5.3

Cancele o fator comum.

$\frac{1}{y^{\frac{1}{2}}} d y = \frac{1}{y (1 + x^{2})} (y x^{4}) d x$

Etapa 3.5.4

Reescreva a expressão.

$\frac{1}{y^{\frac{1}{2}}} d y = \frac{1}{1 + x^{2}} x^{4} d x$

Etapa 3.6

Combine $\frac{1}{1 + x^{2}}$ e $x^{4}$ .

$\frac{1}{y^{\frac{1}{2}}} d y = \frac{x^{4}}{1 + x^{2}} d x$

Etapa 4

Integre os dois lados.

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Etapa 4.1

Determine uma integral de cada lado.

$\int \frac{1}{y^{\frac{1}{2}}} d y = \int \frac{x^{4}}{1 + x^{2}} d x$

Etapa 4.2

Integre o lado esquerdo.

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Etapa 4.2.1

Aplique regras básicas de expoentes.

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Etapa 4.2.1.1

Mova $y^{\frac{1}{2}}$ para fora do denominador, elevando-o à $- 1$ potência.

$\int {(y^{\frac{1}{2}})}^{- 1} d y = \int \frac{x^{4}}{1 + x^{2}} d x$

Etapa 4.2.1.2

Multiplique os expoentes em ${(y^{\frac{1}{2}})}^{- 1}$ .

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Etapa 4.2.1.2.1

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$\int y^{\frac{1}{2} \cdot - 1} d y = \int \frac{x^{4}}{1 + x^{2}} d x$

Etapa 4.2.1.2.2

Combine $\frac{1}{2}$ e $- 1$ .

$\int y^{\frac{- 1}{2}} d y = \int \frac{x^{4}}{1 + x^{2}} d x$

Etapa 4.2.1.2.3

Mova o número negativo para a frente da fração.

$\int y^{- \frac{1}{2}} d y = \int \frac{x^{4}}{1 + x^{2}} d x$

Etapa 4.2.2

De acordo com a regra da multiplicação de potências, a integral de $y^{- \frac{1}{2}}$ com relação a $y$ é $2 y^{\frac{1}{2}}$ .

$2 y^{\frac{1}{2}} + C_{1} = \int \frac{x^{4}}{1 + x^{2}} d x$

Etapa 4.3

Integre o lado direito.

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Etapa 4.3.1

Reordene $1$ e $x^{2}$ .

$2 y^{\frac{1}{2}} + C_{1} = \int \frac{x^{4}}{x^{2} + 1} d x$

Etapa 4.3.2

Divida $x^{4}$ por $x^{2} + 1$ .

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Etapa 4.3.2.1

Estabeleça os polinômios a serem divididos. Se não houver um termo para cada expoente, insira um com valor de $0$ .

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Etapa 4.3.2.2

Divida o termo de ordem mais alta no dividendo $x^{4}$ pelo termo de ordem mais alta no divisor $x^{2}$ .

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Etapa 4.3.2.3

Multiplique o novo termo do quociente pelo divisor.

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

Etapa 4.3.2.4

A expressão precisa ser subtraída do dividendo. Portanto, altere todos os sinais em $x^{4} + 0 + x^{2}$ .

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

Etapa 4.3.2.5

Depois de alterar os sinais, some o último dividendo do polinômio multiplicado para encontrar o novo dividendo.

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

Etapa 4.3.2.6

Tire o próximo termo do dividendo original e o coloque no dividendo atual.

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

$0 x$

$0$

Etapa 4.3.2.7

Divida o termo de ordem mais alta no dividendo $- x^{2}$ pelo termo de ordem mais alta no divisor $x^{2}$ .

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

$0 x$

$0$

Etapa 4.3.2.8

Multiplique o novo termo do quociente pelo divisor.

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

$0 x$

$0$

$x^{2}$

$0$

$1$

Etapa 4.3.2.9

A expressão precisa ser subtraída do dividendo. Portanto, altere todos os sinais em $- x^{2} + 0 - 1$ .

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

$0 x$

$0$

$x^{2}$

$0$

$1$

Etapa 4.3.2.10

Depois de alterar os sinais, some o último dividendo do polinômio multiplicado para encontrar o novo dividendo.

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$x^{2}$

$0 x$

$0$

$x^{2}$

$0$

$1$

Etapa 4.3.2.11

A resposta final é o quociente mais o resto sobre o divisor.

$2 y^{\frac{1}{2}} + C_{1} = \int x^{2} - 1 + \frac{1}{x^{2} + 1} d x$

Etapa 4.3.3

Divida a integral única em várias integrais.

$2 y^{\frac{1}{2}} + C_{1} = \int x^{2} d x + \int - 1 d x + \int \frac{1}{x^{2} + 1} d x$

Etapa 4.3.4

De acordo com a regra da multiplicação de potências, a integral de $x^{2}$ com relação a $x$ é $\frac{1}{3} x^{3}$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{1}{3} x^{3} + C_{2} + \int - 1 d x + \int \frac{1}{x^{2} + 1} d x$

Etapa 4.3.5

Aplique a regra da constante.

$2 y^{\frac{1}{2}} + C_{1} = \frac{1}{3} x^{3} + C_{2} - x + C_{3} + \int \frac{1}{x^{2} + 1} d x$

Etapa 4.3.6

Simplifique a expressão.

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Etapa 4.3.6.1

Reordene $x^{2}$ e $1$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{1}{3} x^{3} + C_{2} - x + C_{3} + \int \frac{1}{1 + x^{2}} d x$

Etapa 4.3.6.2

Reescreva $1$ como $1^{2}$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{1}{3} x^{3} + C_{2} - x + C_{3} + \int \frac{1}{1^{2} + x^{2}} d x$

Etapa 4.3.7

A integral de $\frac{1}{1^{2} + x^{2}}$ com relação a $x$ é $\arctan (x) + C_{4}$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{1}{3} x^{3} + C_{2} - x + C_{3} + \arctan (x) + C_{4}$

Etapa 4.3.8

Simplifique.

$2 y^{\frac{1}{2}} + C_{1} = \frac{1}{3} x^{3} - x + \arctan (x) + C_{5}$

Etapa 4.4

Agrupe a constante de integração no lado direito como $K$ .

$2 y^{\frac{1}{2}} = \frac{1}{3} x^{3} - x + \arctan (x) + K$

Etapa 5

Resolva $y$ .

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Etapa 5.1

Divida cada termo em $2 y^{\frac{1}{2}} = \frac{1}{3} x^{3} - x + \arctan (x) + K$ por $2$ e simplifique.

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Etapa 5.1.1

Divida cada termo em $2 y^{\frac{1}{2}} = \frac{1}{3} x^{3} - x + \arctan (x) + K$ por $2$ .

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{\frac{1}{3} x^{3}}{2} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.2

Simplifique o lado esquerdo.

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Etapa 5.1.2.1

Cancele o fator comum.

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{\frac{1}{3} x^{3}}{2} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.2.2

Divida $y^{\frac{1}{2}}$ por $1$ .

$y^{\frac{1}{2}} = \frac{\frac{1}{3} x^{3}}{2} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.3

Simplifique o lado direito.

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Etapa 5.1.3.1

Simplifique cada termo.

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Etapa 5.1.3.1.1

Combine $\frac{1}{3}$ e $x^{3}$ .

$y^{\frac{1}{2}} = \frac{\frac{x^{3}}{3}}{2} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.3.1.2

Multiplique o numerador pelo inverso do denominador.

$y^{\frac{1}{2}} = \frac{x^{3}}{3} \cdot \frac{1}{2} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.3.1.3

Combine.

$y^{\frac{1}{2}} = \frac{x^{3} \cdot 1}{3 \cdot 2} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.3.1.4

Multiplique $x^{3}$ por $1$ .

$y^{\frac{1}{2}} = \frac{x^{3}}{3 \cdot 2} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.3.1.5

Multiplique $3$ por $2$ .

$y^{\frac{1}{2}} = \frac{x^{3}}{6} + \frac{- x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.1.3.1.6

Mova o número negativo para a frente da fração.

$y^{\frac{1}{2}} = \frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}$

Etapa 5.2

Eleve cada lado da equação à potência de $2$ para eliminar o expoente fracionário no lado esquerdo.

${(y^{\frac{1}{2}})}^{2} = {(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})}^{2}$

Etapa 5.3

Simplifique o expoente.

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Etapa 5.3.1

Simplifique o lado esquerdo.

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Etapa 5.3.1.1

Simplifique ${(y^{\frac{1}{2}})}^{2}$ .

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Etapa 5.3.1.1.1

Multiplique os expoentes em ${(y^{\frac{1}{2}})}^{2}$ .

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Etapa 5.3.1.1.1.1

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{2} \cdot 2} = {(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})}^{2}$

Etapa 5.3.1.1.1.2

Cancele o fator comum de $2$ .

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Etapa 5.3.1.1.1.2.1

Cancele o fator comum.

$y^{\frac{1}{2} \cdot 2} = {(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})}^{2}$

Etapa 5.3.1.1.1.2.2

Reescreva a expressão.

$y^{1} = {(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})}^{2}$

Etapa 5.3.1.1.2

Simplifique.

$y = {(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})}^{2}$

Etapa 5.3.2

Simplifique o lado direito.

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Etapa 5.3.2.1

Simplifique ${(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})}^{2}$ .

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Etapa 5.3.2.1.1

Reescreva ${(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})}^{2}$ como $(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}) (\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})$ .

$y = (\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}) (\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})$

Etapa 5.3.2.1.2

Expanda $(\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2}) (\frac{x^{3}}{6} - \frac{x}{2} + \frac{\arctan (x)}{2} + \frac{K}{2})$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$y = \frac{x^{3}}{6} \cdot \frac{x^{3}}{6} + \frac{x^{3}}{6} (- \frac{x}{2}) + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3

Simplifique os termos.

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Etapa 5.3.2.1.3.1

Simplifique cada termo.

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Etapa 5.3.2.1.3.1.1

Combine.

$y = \frac{x^{3} x^{3}}{6 \cdot 6} + \frac{x^{3}}{6} (- \frac{x}{2}) + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.2

Multiplique $x^{3}$ por $x^{3}$ somando os expoentes.

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Etapa 5.3.2.1.3.1.2.1

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$y = \frac{x^{3 + 3}}{6 \cdot 6} + \frac{x^{3}}{6} (- \frac{x}{2}) + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.2.2

Some $3$ e $3$ .

$y = \frac{x^{6}}{6 \cdot 6} + \frac{x^{3}}{6} (- \frac{x}{2}) + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.3

Multiplique $6$ por $6$ .

$y = \frac{x^{6}}{36} + \frac{x^{3}}{6} (- \frac{x}{2}) + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.4

Reescreva usando a propriedade comutativa da multiplicação.

$y = \frac{x^{6}}{36} - \frac{x^{3}}{6} \cdot \frac{x}{2} + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.5

Multiplique $- \frac{x^{3}}{6} \cdot \frac{x}{2}$ .

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Etapa 5.3.2.1.3.1.5.1

Multiplique $\frac{x}{2}$ por $\frac{x^{3}}{6}$ .

$y = \frac{x^{6}}{36} - \frac{x \cdot x^{3}}{2 \cdot 6} + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.5.2

Multiplique $x$ por $x^{3}$ somando os expoentes.

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Etapa 5.3.2.1.3.1.5.2.1

Multiplique $x$ por $x^{3}$ .

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Etapa 5.3.2.1.3.1.5.2.1.1

Eleve $x$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{1} x^{3}}{2 \cdot 6} + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.5.2.1.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$y = \frac{x^{6}}{36} - \frac{x^{1 + 3}}{2 \cdot 6} + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.5.2.2

Some $1$ e $3$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{2 \cdot 6} + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.5.3

Multiplique $2$ por $6$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3}}{6} \cdot \frac{\arctan (x)}{2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.6

Combine.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{6 \cdot 2} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.7

Multiplique $6$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3}}{6} \cdot \frac{K}{2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.8

Combine.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{6 \cdot 2} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.9

Multiplique $6$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x}{2} \cdot \frac{x^{3}}{6} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.10

Multiplique $- \frac{x}{2} \cdot \frac{x^{3}}{6}$ .

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Etapa 5.3.2.1.3.1.10.1

Multiplique $\frac{x^{3}}{6}$ por $\frac{x}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{3} x}{6 \cdot 2} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.10.2

Multiplique $x^{3}$ por $x$ somando os expoentes.

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Etapa 5.3.2.1.3.1.10.2.1

Multiplique $x^{3}$ por $x$ .

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Etapa 5.3.2.1.3.1.10.2.1.1

Eleve $x$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{3} x^{1}}{6 \cdot 2} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.10.2.1.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{3 + 1}}{6 \cdot 2} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.10.2.2

Some $3$ e $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{6 \cdot 2} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.10.3

Multiplique $6$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} - \frac{x}{2} (- \frac{x}{2}) - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11

Multiplique $- \frac{x}{2} (- \frac{x}{2})$ .

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Etapa 5.3.2.1.3.1.11.1

Multiplique $- 1$ por $- 1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + 1 \frac{x}{2} \frac{x}{2} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11.2

Multiplique $\frac{x}{2}$ por $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x}{2} \cdot \frac{x}{2} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11.3

Multiplique $\frac{x}{2}$ por $\frac{x}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x \cdot x}{2 \cdot 2} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11.4

Eleve $x$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{1} x}{2 \cdot 2} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11.5

Eleve $x$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{1} x^{1}}{2 \cdot 2} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11.6

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{1 + 1}}{2 \cdot 2} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11.7

Some $1$ e $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{2 \cdot 2} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.11.8

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{x}{2} \cdot \frac{\arctan (x)}{2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.12

Multiplique $- \frac{x}{2} \cdot \frac{\arctan (x)}{2}$ .

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Etapa 5.3.2.1.3.1.12.1

Multiplique $\frac{\arctan (x)}{2}$ por $\frac{x}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{2 \cdot 2} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.12.2

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{x}{2} \cdot \frac{K}{2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.13

Multiplique $- \frac{x}{2} \cdot \frac{K}{2}$ .

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Etapa 5.3.2.1.3.1.13.1

Multiplique $\frac{K}{2}$ por $\frac{x}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{2 \cdot 2} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.13.2

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x)}{2} \cdot \frac{x^{3}}{6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.14

Combine.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{2 \cdot 6} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.15

Multiplique $2$ por $6$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} + \frac{\arctan (x)}{2} (- \frac{x}{2}) + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.16

Reescreva usando a propriedade comutativa da multiplicação.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{\arctan (x)}{2} \cdot \frac{x}{2} + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.17

Multiplique $- \frac{\arctan (x)}{2} \cdot \frac{x}{2}$ .

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Etapa 5.3.2.1.3.1.17.1

Multiplique $\frac{x}{2}$ por $\frac{\arctan (x)}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{2 \cdot 2} + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.17.2

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.18

Multiplique $\frac{\arctan (x)}{2} \cdot \frac{\arctan (x)}{2}$ .

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Etapa 5.3.2.1.3.1.18.1

Multiplique $\frac{\arctan (x)}{2}$ por $\frac{\arctan (x)}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{\arctan (x) \arctan (x)}{2 \cdot 2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.18.2

Eleve $\arctan (x)$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{1} \arctan (x)}{2 \cdot 2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.18.3

Eleve $\arctan (x)$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{1} {\arctan (x)}^{1}}{2 \cdot 2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.18.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{1 + 1}}{2 \cdot 2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.18.5

Some $1$ e $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{2 \cdot 2} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.18.6

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x)}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.19

Multiplique $\frac{\arctan (x)}{2} \cdot \frac{K}{2}$ .

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Etapa 5.3.2.1.3.1.19.1

Multiplique $\frac{\arctan (x)}{2}$ por $\frac{K}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{2 \cdot 2} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.19.2

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K}{2} \cdot \frac{x^{3}}{6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.20

Combine.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{2 \cdot 6} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.21

Multiplique $2$ por $6$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} + \frac{K}{2} (- \frac{x}{2}) + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.22

Reescreva usando a propriedade comutativa da multiplicação.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.23

Multiplique $- \frac{K}{2} \cdot \frac{x}{2}$ .

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Etapa 5.3.2.1.3.1.23.1

Multiplique $\frac{x}{2}$ por $\frac{K}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{2 \cdot 2} + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.23.2

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K}{2} \cdot \frac{\arctan (x)}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.24

Multiplique $\frac{K}{2} \cdot \frac{\arctan (x)}{2}$ .

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Etapa 5.3.2.1.3.1.24.1

Multiplique $\frac{K}{2}$ por $\frac{\arctan (x)}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{2 \cdot 2} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.24.2

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Etapa 5.3.2.1.3.1.25

Multiplique $\frac{K}{2} \cdot \frac{K}{2}$ .

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Etapa 5.3.2.1.3.1.25.1

Multiplique $\frac{K}{2}$ por $\frac{K}{2}$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{4} + \frac{K \cdot K}{2 \cdot 2}$

Etapa 5.3.2.1.3.1.25.2

Eleve $K$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{4} + \frac{K^{1} K}{2 \cdot 2}$

Etapa 5.3.2.1.3.1.25.3

Eleve $K$ à potência de $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{4} + \frac{K^{1} K^{1}}{2 \cdot 2}$

Etapa 5.3.2.1.3.1.25.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{4} + \frac{K^{1 + 1}}{2 \cdot 2}$

Etapa 5.3.2.1.3.1.25.5

Some $1$ e $1$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{4} + \frac{K^{2}}{2 \cdot 2}$

Etapa 5.3.2.1.3.1.25.6

Multiplique $2$ por $2$ .

$y = \frac{x^{6}}{36} - \frac{x^{4}}{12} + \frac{x^{3} \arctan (x)}{12} + \frac{x^{3} K}{12} - \frac{x^{4}}{12} + \frac{x^{2}}{4} - \frac{\arctan (x) x}{4} - \frac{K x}{4} + \frac{\arctan (x) x^{3}}{12} - \frac{x \arctan (x)}{4} + \frac{{\arctan (x)}^{2}}{4} + \frac{\arctan (x) K}{4} + \frac{K x^{3}}{12} - \frac{x K}{4} + \frac{K \arctan (x)}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.3.2

Simplifique os termos.

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Etapa 5.3.2.1.3.2.1

Combine os numeradores em relação ao denominador comum.

$y = \frac{x^{6}}{36} + \frac{- x^{4} + x^{3} \arctan (x) + x^{3} K - x^{4} + \arctan (x) x^{3} + K x^{3}}{12} + \frac{x^{2} - \arctan (x) x - K x - x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.3.2.2

Subtraia $x^{4}$ de $- x^{4}$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + x^{3} \arctan (x) + x^{3} K + \arctan (x) x^{3} + K x^{3}}{12} + \frac{x^{2} - \arctan (x) x - K x - x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.4

Some $x^{3} \arctan (x)$ e $\arctan (x) x^{3}$ .

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Etapa 5.3.2.1.4.1

Reordene $\arctan (x)$ e $x^{3}$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + x^{3} \arctan (x) + x^{3} \arctan (x) + x^{3} K + K x^{3}}{12} + \frac{x^{2} - \arctan (x) x - K x - x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.4.2

Some $x^{3} \arctan (x)$ e $x^{3} \arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + x^{3} K + K x^{3}}{12} + \frac{x^{2} - \arctan (x) x - K x - x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.5

Some $x^{3} K$ e $K x^{3}$ .

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Etapa 5.3.2.1.5.1

Reordene $x^{3}$ e $K$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + K x^{3} + K x^{3}}{12} + \frac{x^{2} - \arctan (x) x - K x - x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.5.2

Some $K x^{3}$ e $K x^{3}$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - \arctan (x) x - K x - x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.6

Subtraia $x \arctan (x)$ de $- \arctan (x) x$ .

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Etapa 5.3.2.1.6.1

Mova $\arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - x \arctan (x) - x \arctan (x) - K x + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.6.2

Subtraia $x \arctan (x)$ de $- x \arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - 2 x \arctan (x) - K x + {\arctan (x)}^{2} + \arctan (x) K - x K + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.7

Subtraia $x K$ de $- K x$ .

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Etapa 5.3.2.1.7.1

Mova $x$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - K x - K x + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.7.2

Subtraia $K x$ de $- K x$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + \arctan (x) K - 2 K x + K \arctan (x) + K^{2}}{4}$

Etapa 5.3.2.1.8

Some $\arctan (x) K$ e $K \arctan (x)$ .

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Etapa 5.3.2.1.8.1

Reordene $\arctan (x)$ e $K$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + K \arctan (x) + K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.8.2

Some $K \arctan (x)$ e $K \arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9

Simplifique cada termo.

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Etapa 5.3.2.1.9.1

Fatore $2 x^{3}$ de $- 2 x^{4} + 2 x^{3} \arctan (x) + 2 K x^{3}$ .

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Etapa 5.3.2.1.9.1.1

Fatore $2 x^{3}$ de $- 2 x^{4}$ .

$y = \frac{x^{6}}{36} + \frac{2 x^{3} (- x) + 2 x^{3} \arctan (x) + 2 K x^{3}}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.1.2

Fatore $2 x^{3}$ de $2 x^{3} \arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{2 x^{3} (- x) + 2 x^{3} (\arctan (x)) + 2 K x^{3}}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.1.3

Fatore $2 x^{3}$ de $2 K x^{3}$ .

$y = \frac{x^{6}}{36} + \frac{2 x^{3} (- x) + 2 x^{3} (\arctan (x)) + 2 x^{3} K}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.1.4

Fatore $2 x^{3}$ de $2 x^{3} (- x) + 2 x^{3} (\arctan (x))$ .

$y = \frac{x^{6}}{36} + \frac{2 x^{3} (- x + \arctan (x)) + 2 x^{3} K}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.1.5

Fatore $2 x^{3}$ de $2 x^{3} (- x + \arctan (x)) + 2 x^{3} K$ .

$y = \frac{x^{6}}{36} + \frac{2 x^{3} (- x + \arctan (x) + K)}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.2

Cancele o fator comum de $2$ e $12$ .

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Etapa 5.3.2.1.9.2.1

Fatore $2$ de $2 x^{3} (- x + \arctan (x) + K)$ .

$y = \frac{x^{6}}{36} + \frac{2 (x^{3} (- x + \arctan (x) + K))}{12} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.2.2

Cancele os fatores comuns.

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Etapa 5.3.2.1.9.2.2.1

Fatore $2$ de $12$ .

$y = \frac{x^{6}}{36} + \frac{2 (x^{3} (- x + \arctan (x) + K))}{2 \cdot 6} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.2.2.2

Cancele o fator comum.

$y = \frac{x^{6}}{36} + \frac{2 (x^{3} (- x + \arctan (x) + K))}{2 \cdot 6} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.2.2.3

Reescreva a expressão.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$

Etapa 5.3.2.1.9.3

Divida a fração $\frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x + K^{2}}{4}$ em duas frações.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4

Simplifique cada termo.

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Etapa 5.3.2.1.9.4.1

Divida a fração $\frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x) - 2 K x}{4}$ em duas frações.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x)}{4} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2

Simplifique cada termo.

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Etapa 5.3.2.1.9.4.2.1

Divida a fração $\frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2} + 2 K \arctan (x)}{4}$ em duas frações.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{x^{2} - 2 x \arctan (x) + {\arctan (x)}^{2}}{4} + \frac{2 K \arctan (x)}{4} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.2

Simplifique cada termo.

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Etapa 5.3.2.1.9.4.2.2.1

Fatore usando a regra do quadrado perfeito.

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Etapa 5.3.2.1.9.4.2.2.1.1

Verifique se o termo do meio é duas vezes o produto dos números ao quadrado no primeiro e no terceiro termos.

$2 x \arctan (x) = 2 \cdot x \cdot \arctan (x)$

Etapa 5.3.2.1.9.4.2.2.1.2

Reescreva o polinômio.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{x^{2} - 2 \cdot x \cdot \arctan (x) + {\arctan (x)}^{2}}{4} + \frac{2 K \arctan (x)}{4} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.2.1.3

Fatore usando a regra do trinômio quadrado perfeito $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , em que $a = x$ e $b = \arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{2 K \arctan (x)}{4} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.2.2

Cancele o fator comum de $2$ e $4$ .

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Etapa 5.3.2.1.9.4.2.2.2.1

Fatore $2$ de $2 K \arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{2 (K \arctan (x))}{4} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.2.2.2

Cancele os fatores comuns.

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Etapa 5.3.2.1.9.4.2.2.2.2.1

Fatore $2$ de $4$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{2 (K \arctan (x))}{2 \cdot 2} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.2.2.2.2

Cancele o fator comum.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{2 (K \arctan (x))}{2 \cdot 2} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.2.2.2.3

Reescreva a expressão.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} + \frac{- 2 K x}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.3

Cancele o fator comum de $- 2$ e $4$ .

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Etapa 5.3.2.1.9.4.2.3.1

Fatore $2$ de $- 2 K x$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} + \frac{2 (- K x)}{4} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.3.2

Cancele os fatores comuns.

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Etapa 5.3.2.1.9.4.2.3.2.1

Fatore $2$ de $4$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} + \frac{2 (- K x)}{2 (2)} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.3.2.2

Cancele o fator comum.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} + \frac{2 (- K x)}{2 \cdot 2} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.3.2.3

Reescreva a expressão.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} + \frac{- K x}{2} + \frac{K^{2}}{4}$

Etapa 5.3.2.1.9.4.2.4

Mova o número negativo para a frente da fração.

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} - \frac{K x}{2} + \frac{K^{2}}{4}$

Etapa 5.4

Simplifique $\frac{x^{6}}{36} + \frac{x^{3} (- x + \arctan (x) + K)}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} - \frac{K x}{2} + \frac{K^{2}}{4}$ .

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Etapa 5.4.1

Mova $\arctan (x)$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + K + \arctan (x))}{6} + \frac{{(x - \arctan (x))}^{2}}{4} + \frac{K \arctan (x)}{2} - \frac{K x}{2} + \frac{K^{2}}{4}$

Etapa 5.4.2

Mova $\frac{{(x - \arctan (x))}^{2}}{4}$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + K + \arctan (x))}{6} + \frac{K \arctan (x)}{2} - \frac{K x}{2} + \frac{K^{2}}{4} + \frac{{(x - \arctan (x))}^{2}}{4}$

Etapa 5.4.3

Mova $\frac{K \arctan (x)}{2}$ .

$y = \frac{x^{6}}{36} + \frac{x^{3} (- x + K + \arctan (x))}{6} - \frac{K x}{2} + \frac{K^{2}}{4} + \frac{K \arctan (x)}{2} + \frac{{(x - \arctan (x))}^{2}}{4}$

Etapa 5.4.4

Mova $\frac{x^{3} (- x + K + \arctan (x))}{6}$ .

$y = \frac{x^{6}}{36} - \frac{K x}{2} + \frac{K^{2}}{4} + \frac{x^{3} (- x + K + \arctan (x))}{6} + \frac{K \arctan (x)}{2} + \frac{{(x - \arctan (x))}^{2}}{4}$

Etapa 6

Simplifique a constante de integração.

$y = \frac{x^{6}}{36} - \frac{K x}{2} + K + \frac{x^{3} (- x + K + \arctan (x))}{6} + \frac{K \arctan (x)}{2} + \frac{{(x - \arctan (x))}^{2}}{4}$