Calculus Examples

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} + u^{4} t^{2}}$

Step 1

Separate the variables.

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Step 1.1

Factor.

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Step 1.1.1

Factor $u t^{2}$ out of $u t^{2} + u^{4} t^{2}$ .

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Step 1.1.1.1

Factor $u t^{2}$ out of $u t^{2}$ .

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} (1) + u^{4} t^{2}}$

Step 1.1.1.2

Factor $u t^{2}$ out of $u^{4} t^{2}$ .

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} (1) + u t^{2} (u^{3})}$

Step 1.1.1.3

Factor $u t^{2}$ out of $u t^{2} (1) + u t^{2} (u^{3})$ .

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} (1 + u^{3})}$

Step 1.1.2

Rewrite $1$ as $1^{3}$ .

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} (1^{3} + u^{3})}$

Step 1.1.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = u$ .

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} ((1 + u) (1^{2} - 1 u + u^{2}))}$

Step 1.1.4

Factor.

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Step 1.1.4.1

Simplify.

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Step 1.1.4.1.1

One to any power is one.

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} ((1 + u) (1 - 1 u + u^{2}))}$

Step 1.1.4.1.2

Rewrite $- 1 u$ as $- u$ .

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} ((1 + u) (1 - u + u^{2}))}$

Step 1.1.4.2

Remove unnecessary parentheses.

$\frac{d u}{d t} = \frac{1 + t^{4}}{u t^{2} (1 + u) (1 - u + u^{2})}$

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $u (1 + u) (1 - u + u^{2})$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

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

Step 1.4.2

Multiply $u$ by $1$ .

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

Step 1.4.3

Multiply $u$ by $u$ .

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

Step 1.4.4

Expand $(u + u^{2}) (1 - u + u^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.4.5

Combine the opposite terms in $u \cdot 1 + u (- u) + u \cdot u^{2} + u^{2} \cdot 1 + u^{2} (- u) + u^{2} u^{2}$ .

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Step 1.4.5.1

Reorder the factors in the terms $u \cdot u^{2}$ and $u^{2} (- u)$ .

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

Step 1.4.5.2

Subtract $u^{2} u$ from $u^{2} u$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = (u \cdot 1 + u (- u) + u^{2} \cdot 1 + 0 + u^{2} u^{2}) (\frac{1 + t^{4}}{t^{2}} \cdot \frac{1}{u (1 + u) (1 - u + u^{2})})$

Step 1.4.5.3

Add $u \cdot 1 + u (- u) + u^{2} \cdot 1$ and $0$ .

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

Step 1.4.6

Simplify each term.

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Step 1.4.6.1

Multiply $u$ by $1$ .

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

Step 1.4.6.2

Rewrite using the commutative property of multiplication.

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

Step 1.4.6.3

Multiply $u$ by $u$ by adding the exponents.

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Step 1.4.6.3.1

Move $u$ .

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

Step 1.4.6.3.2

Multiply $u$ by $u$ .

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

Step 1.4.6.4

Multiply $u^{2}$ by $1$ .

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

Step 1.4.6.5

Multiply $u^{2}$ by $u^{2}$ by adding the exponents.

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Step 1.4.6.5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.4.6.5.2

Add $2$ and $2$ .

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

Step 1.4.7

Combine the opposite terms in $u - u^{2} + u^{2} + u^{4}$ .

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Step 1.4.7.1

Add $- u^{2}$ and $u^{2}$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = (u + 0 + u^{4}) (\frac{1 + t^{4}}{t^{2}} \cdot \frac{1}{u (1 + u) (1 - u + u^{2})})$

Step 1.4.7.2

Add $u$ and $0$ .

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

Step 1.4.8

Multiply $\frac{1 + t^{4}}{t^{2}}$ by $\frac{1}{u (1 + u) (1 - u + u^{2})}$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = (u + u^{4}) \frac{1 + t^{4}}{t^{2} (u (1 + u) (1 - u + u^{2}))}$

Step 1.4.9

Remove parentheses.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = (u + u^{4}) \frac{1 + t^{4}}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.10

Multiply $u + u^{4}$ by $\frac{1 + t^{4}}{t^{2} u (1 + u) (1 - u + u^{2})}$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{(u + u^{4}) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.11

Simplify the numerator.

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Step 1.4.11.1

Factor $u$ out of $u + u^{4}$ .

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Step 1.4.11.1.1

Raise $u$ to the power of $1$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{(u^{1} + u^{4}) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.11.1.2

Factor $u$ out of $u^{1}$ .

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

Step 1.4.11.1.3

Factor $u$ out of $u^{4}$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{(u \cdot 1 + u \cdot u^{3}) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.11.1.4

Factor $u$ out of $u \cdot 1 + u \cdot u^{3}$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{u (1 + u^{3}) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.11.2

Rewrite $1$ as $1^{3}$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{u (1^{3} + u^{3}) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.11.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = u$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{u ((1 + u) (1^{2} - 1 u + u^{2})) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.11.4

Simplify.

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Step 1.4.11.4.1

One to any power is one.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{u ((1 + u) (1 - 1 u + u^{2})) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.11.4.2

Rewrite $- 1 u$ as $- u$ .

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{u ((1 + u) (1 - u + u^{2})) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{u (1 + u) (1 - u + u^{2}) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.12

Cancel the common factor of $u$ .

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Step 1.4.12.1

Cancel the common factor.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{u (1 + u) (1 - u + u^{2}) (1 + t^{4})}{t^{2} u (1 + u) (1 - u + u^{2})}$

Step 1.4.12.2

Rewrite the expression.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{((1 + u) (1 - u + u^{2})) (1 + t^{4})}{(t^{2} (1 + u)) (1 - u + u^{2})}$

Step 1.4.13

Cancel the common factor of $1 + u$ .

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Step 1.4.13.1

Cancel the common factor.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{(1 + u) (1 - u + u^{2}) (1 + t^{4})}{t^{2} (1 + u) (1 - u + u^{2})}$

Step 1.4.13.2

Rewrite the expression.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{(1 - u + u^{2}) (1 + t^{4})}{(t^{2}) (1 - u + u^{2})}$

Step 1.4.14

Cancel the common factor of $1 - u + u^{2}$ .

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Step 1.4.14.1

Cancel the common factor.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{(1 - u + u^{2}) (1 + t^{4})}{t^{2} (1 - u + u^{2})}$

Step 1.4.14.2

Rewrite the expression.

$u (1 + u) (1 - u + u^{2}) \frac{d u}{d t} = \frac{1 + t^{4}}{t^{2}}$

Step 1.5

Rewrite the equation.

$u (1 + u) (1 - u + u^{2}) d u = \frac{1 + t^{4}}{t^{2}} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Apply the distributive property.

$\int (u \cdot 1 + u \cdot u) (1 - u + u^{2}) d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.2

Apply the distributive property.

$\int u \cdot 1 (1 - u + u^{2}) + u \cdot u (1 - u + u^{2}) d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.3

Apply the distributive property.

$\int u \cdot 1 (1 - u) + u \cdot 1 u^{2} + u \cdot u (1 - u + u^{2}) d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.4

Apply the distributive property.

$\int u \cdot 1 \cdot 1 + u \cdot 1 (- u) + u \cdot 1 u^{2} + u \cdot u (1 - u + u^{2}) d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.5

Apply the distributive property.

$\int u \cdot 1 \cdot 1 + u \cdot 1 (- u) + u \cdot 1 u^{2} + u \cdot u (1 - u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.6

Apply the distributive property.

$\int u \cdot 1 \cdot 1 + u \cdot 1 (- u) + u \cdot 1 u^{2} + u \cdot u \cdot 1 + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.7

Reorder $u$ and $1$ .

$\int 1 \cdot u \cdot 1 + u \cdot 1 (- u) + u \cdot 1 u^{2} + u \cdot u \cdot 1 + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.8

Move $u$ .

$\int 1 \cdot 1 u + u \cdot 1 (- u) + u \cdot 1 u^{2} + u \cdot u \cdot 1 + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.9

Reorder $u$ and $1$ .

$\int 1 \cdot 1 u + 1 \cdot u (- u) + u \cdot 1 u^{2} + u \cdot u \cdot 1 + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.10

Move $u$ .

$\int 1 \cdot 1 u + 1 \cdot - 1 u \cdot u + u \cdot 1 u^{2} + u \cdot u \cdot 1 + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.11

Reorder $u$ and $1$ .

$\int 1 \cdot 1 u + 1 \cdot - 1 u \cdot u + 1 \cdot u \cdot u^{2} + u \cdot u \cdot 1 + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.12

Move $u$ .

$\int 1 \cdot 1 u + 1 \cdot - 1 u \cdot u + 1 \cdot u \cdot u^{2} + u \cdot 1 u + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.13

Reorder $u$ and $1$ .

$\int 1 \cdot 1 u + 1 \cdot - 1 u \cdot u + 1 \cdot u \cdot u^{2} + 1 \cdot u \cdot u + u \cdot u (- u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.14

Move $u$ .

$\int 1 \cdot 1 u + 1 \cdot - 1 u \cdot u + 1 \cdot u \cdot u^{2} + 1 \cdot u \cdot u + u \cdot - 1 u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.15

Reorder $u$ and $- 1$ .

$\int 1 \cdot 1 u + 1 \cdot - 1 u \cdot u + 1 \cdot u \cdot u^{2} + 1 \cdot u \cdot u - 1 \cdot u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.16

Multiply $1$ by $1$ .

$\int 1 u + 1 \cdot - 1 u \cdot u + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.17

Multiply $u$ by $1$ .

$\int u + 1 \cdot - 1 u \cdot u + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.18

Multiply $- 1$ by $1$ .

$\int u - u \cdot u + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.19

Factor out negative.

$\int u - (u \cdot u) + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.20

Raise $u$ to the power of $1$ .

$\int u - (u^{1} u) + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.21

Raise $u$ to the power of $1$ .

$\int u - (u^{1} u^{1}) + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.22

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u - u^{1 + 1} + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.23

Add $1$ and $1$ .

$\int u - u^{2} + 1 u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.24

Multiply $u$ by $1$ .

$\int u - u^{2} + u \cdot u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.25

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{1} u^{2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.26

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u - u^{2} + u^{1 + 2} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.27

Add $1$ and $2$ .

$\int u - u^{2} + u^{3} + 1 u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.28

Multiply $u$ by $1$ .

$\int u - u^{2} + u^{3} + u \cdot u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.29

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{3} + u^{1} u - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.30

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{3} + u^{1} u^{1} - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.31

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u - u^{2} + u^{3} + u^{1 + 1} - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.32

Add $1$ and $1$ .

$\int u - u^{2} + u^{3} + u^{2} - 1 u \cdot u \cdot u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.33

Factor out negative.

$\int u - u^{2} + u^{3} + u^{2} - (u \cdot u) u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.34

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{3} + u^{2} - (u^{1} u) u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.35

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{3} + u^{2} - (u^{1} u^{1}) u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.36

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u - u^{2} + u^{3} + u^{2} - u^{1 + 1} u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.37

Add $1$ and $1$ .

$\int u - u^{2} + u^{3} + u^{2} - u^{2} u + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.38

Factor out negative.

$\int u - u^{2} + u^{3} + u^{2} - (u^{2} u) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.39

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{3} + u^{2} - (u^{2} u^{1}) + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.40

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u - u^{2} + u^{3} + u^{2} - u^{2 + 1} + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.41

Add $2$ and $1$ .

$\int u - u^{2} + u^{3} + u^{2} - u^{3} + u \cdot u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.42

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{3} + u^{2} - u^{3} + u^{1} u \cdot u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.43

Raise $u$ to the power of $1$ .

$\int u - u^{2} + u^{3} + u^{2} - u^{3} + u^{1} u^{1} u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.44

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u - u^{2} + u^{3} + u^{2} - u^{3} + u^{1 + 1} u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.45

Add $1$ and $1$ .

$\int u - u^{2} + u^{3} + u^{2} - u^{3} + u^{2} u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.46

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u - u^{2} + u^{3} + u^{2} - u^{3} + u^{2 + 2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.47

Add $2$ and $2$ .

$\int u - u^{2} + u^{3} + u^{2} - u^{3} + u^{4} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.48

Reorder $u$ and $- u^{2}$ .

$\int - u^{2} + u + u^{3} + u^{2} - u^{3} + u^{4} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.49

Move $u$ .

$\int - u^{2} + u^{3} + u + u^{2} - u^{3} + u^{4} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.50

Reorder $- u^{2}$ and $u^{3}$ .

$\int u^{3} - u^{2} + u + u^{2} - u^{3} + u^{4} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.51

Reorder $u^{2}$ and $- u^{3}$ .

$\int u^{3} - u^{2} + u - u^{3} + u^{2} + u^{4} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.52

Move $u^{2}$ .

$\int u^{3} - u^{2} + u - u^{3} + u^{4} + u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.53

Reorder $- u^{3}$ and $u^{4}$ .

$\int u^{3} - u^{2} + u + u^{4} - u^{3} + u^{2} d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.54

Move $u$ .

$\int u^{3} - u^{2} + u^{4} - u^{3} + u^{2} + u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.55

Move $- u^{2}$ .

$\int u^{3} + u^{4} - u^{3} + u^{2} - u^{2} + u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.56

Reorder $u^{3}$ and $u^{4}$ .

$\int u^{4} + u^{3} - u^{3} + u^{2} - u^{2} + u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.57

Subtract $u^{3}$ from $u^{3}$ .

$\int u^{4} + 0 + u^{2} - u^{2} + u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.58

Add $u^{4}$ and $0$ .

$\int u^{4} + u^{2} - u^{2} + u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.59

Subtract $u^{2}$ from $u^{2}$ .

$\int u^{4} + 0 + u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.1.60

Add $u^{4}$ and $0$ .

$\int u^{4} + u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.2

Split the single integral into multiple integrals.

$\int u^{4} d u + \int u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.3

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{5} u^{5} + C_{1} + \int u d u = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.4

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$\frac{1}{5} u^{5} + C_{1} + \frac{1}{2} u^{2} + C_{2} = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.2.5

Simplify.

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int \frac{1 + t^{4}}{t^{2}} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

Move $t^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int (1 + t^{4}) {(t^{2})}^{- 1} d t$

Step 2.3.1.2

Multiply the exponents in ${(t^{2})}^{- 1}$ .

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Step 2.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int (1 + t^{4}) t^{2 \cdot - 1} d t$

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int (1 + t^{4}) t^{- 2} d t$

Step 2.3.2

Multiply $(1 + t^{4}) t^{- 2}$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int 1 t^{- 2} + t^{4} t^{- 2} d t$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $t^{- 2}$ by $1$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int t^{- 2} + t^{4} t^{- 2} d t$

Step 2.3.3.2

Multiply $t^{4}$ by $t^{- 2}$ by adding the exponents.

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Step 2.3.3.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int t^{- 2} + t^{4 - 2} d t$

Step 2.3.3.2.2

Subtract $2$ from $4$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int t^{- 2} + t^{2} d t$

Step 2.3.4

Split the single integral into multiple integrals.

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \int t^{- 2} d t + \int t^{2} d t$

Step 2.3.5

By the Power Rule, the integral of $t^{- 2}$ with respect to $t$ is $- t^{- 1}$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = - t^{- 1} + C_{4} + \int t^{2} d t$

Step 2.3.6

By the Power Rule, the integral of $t^{2}$ with respect to $t$ is $\frac{1}{3} t^{3}$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = - t^{- 1} + C_{4} + \frac{1}{3} t^{3} + C_{5}$

Step 2.3.7

Simplify.

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = - \frac{1}{t} + \frac{1}{3} t^{3} + C_{6}$

Step 2.3.8

Reorder terms.

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} + C_{3} = \frac{1}{3} t^{3} - \frac{1}{t} + C_{6}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{5} u^{5} + \frac{1}{2} u^{2} = \frac{1}{3} t^{3} - \frac{1}{t} + K$