Calculus Examples

$\int \frac{1}{5 + 2 x^{6}} (12 x^{5}) d x$

Step 1

Simplify.

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Step 1.1

Combine $12$ and $\frac{1}{5 + 2 x^{6}}$ .

$\int \frac{12}{5 + 2 x^{6}} \cdot x^{5} d x$

Step 1.2

Combine $\frac{12}{5 + 2 x^{6}}$ and $x^{5}$ .

$\int \frac{12 x^{5}}{5 + 2 x^{6}} d x$

Step 2

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int \frac{x^{5}}{5 + 2 x^{6}} d x$

Step 3

Rewrite $x^{6}$ as ${(x^{6})}^{1}$ .

$12 \int \frac{x^{5}}{5 + 2 {(x^{6})}^{1}} d x$

Step 4

Let $u_{1} = x^{6}$ . Then $d u_{1} = 6 x^{5} d x$ , so $\frac{1}{6} d u_{1} = x^{5} d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.1

Let $u_{1} = x^{6}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 4.1.1

Differentiate $x^{6}$ .

$\frac{d}{d x} [x^{6}]$

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$6 x^{5}$

Step 4.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$12 \int \frac{1}{5 + 2 {u_{1}}^{1}} \cdot \frac{1}{6} d u_{1}$

Step 5

Simplify.

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Step 5.1

Simplify.

$12 \int \frac{1}{5 + 2 u_{1}} \cdot \frac{1}{6} d u_{1}$

Step 5.2

Multiply $\frac{1}{5 + 2 u_{1}}$ by $\frac{1}{6}$ .

$12 \int \frac{1}{(5 + 2 u_{1}) \cdot 6} d u_{1}$

Step 5.3

Move $6$ to the left of $5 + 2 u_{1}$ .

$12 \int \frac{1}{6 (5 + 2 u_{1})} d u_{1}$

Step 6

Since $\frac{1}{6}$ is constant with respect to $u_{1}$ , move $\frac{1}{6}$ out of the integral.

$12 (\frac{1}{6} \int \frac{1}{5 + 2 u_{1}} d u_{1})$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{6}$ and $12$ .

$\frac{12}{6} \int \frac{1}{5 + 2 u_{1}} d u_{1}$

Step 7.2

Cancel the common factor of $12$ and $6$ .

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Step 7.2.1

Factor $6$ out of $12$ .

$\frac{6 \cdot 2}{6} \int \frac{1}{5 + 2 u_{1}} d u_{1}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $6$ out of $6$ .

$\frac{6 \cdot 2}{6 (1)} \int \frac{1}{5 + 2 u_{1}} d u_{1}$

Step 7.2.2.2

Cancel the common factor.

$\frac{6 \cdot 2}{6 \cdot 1} \int \frac{1}{5 + 2 u_{1}} d u_{1}$

Step 7.2.2.3

Rewrite the expression.

$\frac{2}{1} \int \frac{1}{5 + 2 u_{1}} d u_{1}$

Step 7.2.2.4

Divide $2$ by $1$ .

$2 \int \frac{1}{5 + 2 u_{1}} d u_{1}$

Step 8

Let $u_{2} = 5 + 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 8.1

Let $u_{2} = 5 + 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 8.1.1

Differentiate $5 + 2 u_{1}$ .

$\frac{d}{d u_{1}} [5 + 2 u_{1}]$

Step 8.1.2

Differentiate.

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Step 8.1.2.1

By the Sum Rule, the derivative of $5 + 2 u_{1}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [5] + \frac{d}{d u_{1}} [2 u_{1}]$ .

$\frac{d}{d u_{1}} [5] + \frac{d}{d u_{1}} [2 u_{1}]$

Step 8.1.2.2

Since $5$ is constant with respect to $u_{1}$ , the derivative of $5$ with respect to $u_{1}$ is $0$ .

$0 + \frac{d}{d u_{1}} [2 u_{1}]$

Step 8.1.3

Evaluate $\frac{d}{d u_{1}} [2 u_{1}]$ .

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Step 8.1.3.1

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$0 + 2 \frac{d}{d u_{1}} [u_{1}]$

Step 8.1.3.2

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

$0 + 2 \cdot 1$

Step 8.1.3.3

Multiply $2$ by $1$ .

$0 + 2$

Step 8.1.4

Add $0$ and $2$ .

$2$

Step 8.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$2 \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 9

Simplify.

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Step 9.1

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

$2 \int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 9.2

Move $2$ to the left of $u_{2}$ .

$2 \int \frac{1}{2 u_{2}} d u_{2}$

Step 10

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

$2 (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2})$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{2}$ and $2$ .

$\frac{2}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 11.2

Cancel the common factor of $2$ .

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Step 11.2.1

Cancel the common factor.

$\frac{2}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 11.2.2

Rewrite the expression.

$1 \int \frac{1}{u_{2}} d u_{2}$

Step 11.3

Multiply $\int \frac{1}{u_{2}} d u_{2}$ by $1$ .

$\int \frac{1}{u_{2}} d u_{2}$

Step 12

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| u_{2} |) + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{2}$ with $5 + 2 u_{1}$ .

$\ln (| 5 + 2 u_{1} |) + C$

Step 13.2

Replace all occurrences of $u_{1}$ with $x^{6}$ .

$\ln (| 5 + 2 x^{6} |) + C$