Calculus Examples

$\frac{d y}{d x} = x {(2 + x^{2})}^{4}$ , $y (0) = 0$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 2 + x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 2 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $2 + x^{2}$ .

$\frac{d}{d x} [2 + x^{2}]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $2 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [x^{2}]$

Step 2.3.1.1.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 2.3.1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 + 2 x$

Step 2.3.1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 2.3.1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.3.2

Combine $u^{4}$ and $\frac{1}{2}$ .

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

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.4

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

$y + C_{1} = \frac{1}{2} (\frac{1}{5} u^{5} + C_{2})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Rewrite $\frac{1}{2} (\frac{1}{5} u^{5} + C_{2})$ as $\frac{1}{2} \cdot \frac{1}{5} u^{5} + C_{2}$ .

$y + C_{1} = \frac{1}{2} \cdot \frac{1}{5} u^{5} + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$y + C_{1} = \frac{1}{2 \cdot 5} u^{5} + C_{2}$

Step 2.3.5.2.2

Multiply $2$ by $5$ .

$y + C_{1} = \frac{1}{10} u^{5} + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $2 + x^{2}$ .

$y + C_{1} = \frac{1}{10} {(2 + x^{2})}^{5} + C_{2}$

Step 2.4

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

$y = \frac{1}{10} {(2 + x^{2})}^{5} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $0$ for $y$ in $y = \frac{1}{10} {(2 + x^{2})}^{5} + K$ .

$0 = \frac{1}{10} {(2 + 0^{2})}^{5} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{10} \cdot {(2 + 0^{2})}^{5} + K = 0$ .

$\frac{1}{10} \cdot {(2 + 0^{2})}^{5} + K = 0$

Step 4.2

Simplify each term.

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Step 4.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{10} \cdot {(2 + 0)}^{5} + K = 0$

Step 4.2.2

Add $2$ and $0$ .

$\frac{1}{10} \cdot 2^{5} + K = 0$

Step 4.2.3

Raise $2$ to the power of $5$ .

$\frac{1}{10} \cdot 32 + K = 0$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Factor $2$ out of $10$ .

$\frac{1}{2 (5)} \cdot 32 + K = 0$

Step 4.2.4.2

Factor $2$ out of $32$ .

$\frac{1}{2 \cdot 5} \cdot (2 \cdot 16) + K = 0$

Step 4.2.4.3

Cancel the common factor.

$\frac{1}{2 \cdot 5} \cdot (2 \cdot 16) + K = 0$

Step 4.2.4.4

Rewrite the expression.

$\frac{1}{5} \cdot 16 + K = 0$

Step 4.2.5

Combine $\frac{1}{5}$ and $16$ .

$\frac{16}{5} + K = 0$

Step 4.3

Subtract $\frac{16}{5}$ from both sides of the equation.

$K = - \frac{16}{5}$

Step 5

Substitute $- \frac{16}{5}$ for $K$ in $y = \frac{1}{10} {(2 + x^{2})}^{5} + K$ and simplify.

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Step 5.1

Substitute $- \frac{16}{5}$ for $K$ .

$y = \frac{1}{10} {(2 + x^{2})}^{5} - \frac{16}{5}$

Step 5.2

Simplify each term.

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Step 5.2.1

Use the Binomial Theorem.

$y = \frac{1}{10} (2^{5} + 5 \cdot 2^{4} x^{2} + 10 \cdot 2^{3} {(x^{2})}^{2} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Raise $2$ to the power of $5$ .

$y = \frac{1}{10} (32 + 5 \cdot 2^{4} x^{2} + 10 \cdot 2^{3} {(x^{2})}^{2} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.2

Raise $2$ to the power of $4$ .

$y = \frac{1}{10} (32 + 5 \cdot 16 x^{2} + 10 \cdot 2^{3} {(x^{2})}^{2} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.3

Multiply $5$ by $16$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 10 \cdot 2^{3} {(x^{2})}^{2} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.4

Raise $2$ to the power of $3$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 10 \cdot 8 {(x^{2})}^{2} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.5

Multiply $10$ by $8$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 {(x^{2})}^{2} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.6

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 5.2.2.6.1

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

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{2 \cdot 2} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.6.2

Multiply $2$ by $2$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 10 \cdot 2^{2} {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.7

Raise $2$ to the power of $2$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 10 \cdot 4 {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.8

Multiply $10$ by $4$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 {(x^{2})}^{3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.9

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 5.2.2.9.1

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

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 x^{2 \cdot 3} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.9.2

Multiply $2$ by $3$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 x^{6} + 5 \cdot 2 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.10

Multiply $5$ by $2$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 x^{6} + 10 {(x^{2})}^{4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.11

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 5.2.2.11.1

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

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 x^{6} + 10 x^{2 \cdot 4} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.11.2

Multiply $2$ by $4$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 x^{6} + 10 x^{8} + {(x^{2})}^{5}) - \frac{16}{5}$

Step 5.2.2.12

Multiply the exponents in ${(x^{2})}^{5}$ .

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Step 5.2.2.12.1

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

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 x^{6} + 10 x^{8} + x^{2 \cdot 5}) - \frac{16}{5}$

Step 5.2.2.12.2

Multiply $2$ by $5$ .

$y = \frac{1}{10} (32 + 80 x^{2} + 80 x^{4} + 40 x^{6} + 10 x^{8} + x^{10}) - \frac{16}{5}$

Step 5.2.3

Apply the distributive property.

$y = \frac{1}{10} \cdot 32 + \frac{1}{10} (80 x^{2}) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4

Simplify.

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Step 5.2.4.1

Cancel the common factor of $2$ .

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Step 5.2.4.1.1

Factor $2$ out of $10$ .

$y = \frac{1}{2 (5)} \cdot 32 + \frac{1}{10} (80 x^{2}) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.1.2

Factor $2$ out of $32$ .

$y = \frac{1}{2 \cdot 5} \cdot (2 \cdot 16) + \frac{1}{10} (80 x^{2}) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.1.3

Cancel the common factor.

$y = \frac{1}{2 \cdot 5} \cdot (2 \cdot 16) + \frac{1}{10} (80 x^{2}) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.1.4

Rewrite the expression.

$y = \frac{1}{5} \cdot 16 + \frac{1}{10} (80 x^{2}) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.2

Combine $\frac{1}{5}$ and $16$ .

$y = \frac{16}{5} + \frac{1}{10} (80 x^{2}) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.3

Cancel the common factor of $10$ .

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Step 5.2.4.3.1

Factor $10$ out of $80 x^{2}$ .

$y = \frac{16}{5} + \frac{1}{10} (10 (8 x^{2})) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.3.2

Cancel the common factor.

$y = \frac{16}{5} + \frac{1}{10} (10 (8 x^{2})) + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.3.3

Rewrite the expression.

$y = \frac{16}{5} + 8 x^{2} + \frac{1}{10} (80 x^{4}) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.4

Cancel the common factor of $10$ .

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Step 5.2.4.4.1

Factor $10$ out of $80 x^{4}$ .

$y = \frac{16}{5} + 8 x^{2} + \frac{1}{10} (10 (8 x^{4})) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.4.2

Cancel the common factor.

$y = \frac{16}{5} + 8 x^{2} + \frac{1}{10} (10 (8 x^{4})) + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.4.3

Rewrite the expression.

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + \frac{1}{10} (40 x^{6}) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.5

Cancel the common factor of $10$ .

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Step 5.2.4.5.1

Factor $10$ out of $40 x^{6}$ .

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + \frac{1}{10} (10 (4 x^{6})) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.5.2

Cancel the common factor.

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + \frac{1}{10} (10 (4 x^{6})) + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.5.3

Rewrite the expression.

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + 4 x^{6} + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.6

Cancel the common factor of $10$ .

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Step 5.2.4.6.1

Factor $10$ out of $10 x^{8}$ .

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + 4 x^{6} + \frac{1}{10} (10 (x^{8})) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.6.2

Cancel the common factor.

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + 4 x^{6} + \frac{1}{10} (10 x^{8}) + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.6.3

Rewrite the expression.

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{1}{10} x^{10} - \frac{16}{5}$

Step 5.2.4.7

Combine $\frac{1}{10}$ and $x^{10}$ .

$y = \frac{16}{5} + 8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{x^{10}}{10} - \frac{16}{5}$

Step 5.3

Combine the opposite terms in $\frac{16}{5} + 8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{x^{10}}{10} - \frac{16}{5}$ .

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Step 5.3.1

Combine the numerators over the common denominator.

$y = 8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{x^{10}}{10} + \frac{16 - 16}{5}$

Step 5.3.2

Subtract $16$ from $16$ .

$y = 8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{x^{10}}{10} + \frac{0}{5}$

Step 5.3.3

Divide $0$ by $5$ .

$y = 8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{x^{10}}{10} + 0$

Step 5.3.4

Add $8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{x^{10}}{10}$ and $0$ .

$y = 8 x^{2} + 8 x^{4} + 4 x^{6} + x^{8} + \frac{x^{10}}{10}$