Calculus Examples

$\int 6 x^{3} {(1 + x^{8})}^{- 1} d x$

Step 1

Simplify.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int 6 x^{3} \frac{1}{1 + x^{8}} d x$

Step 1.2

Multiply $6 x^{3} \frac{1}{1 + x^{8}}$ .

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Step 1.2.1

Combine $6$ and $\frac{1}{1 + x^{8}}$ .

$\int x^{3} \frac{6}{1 + x^{8}} d x$

Step 1.2.2

Combine $x^{3}$ and $\frac{6}{1 + x^{8}}$ .

$\int \frac{x^{3} \cdot 6}{1 + x^{8}} d x$

Step 1.3

Move $6$ to the left of $x^{3}$ .

$\int \frac{6 x^{3}}{1 + x^{8}} d x$

Step 2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{x^{3}}{1 + x^{8}} d x$

Step 3

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$6 \int \frac{x^{3}}{1 + {(x^{4})}^{2}} d x$

Step 4

Let $u = x^{4}$ . Then $d u = 4 x^{3} d x$ , so $\frac{1}{4} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x^{4}$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x^{4}$ .

$\frac{d}{d x} [x^{4}]$

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3}$

Step 4.2

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

$6 \int \frac{1}{1 + u^{2}} \cdot \frac{1}{4} d u$

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1}{1 + u^{2}}$ by $\frac{1}{4}$ .

$6 \int \frac{1}{(1 + u^{2}) \cdot 4} d u$

Step 5.2

Move $4$ to the left of $1 + u^{2}$ .

$6 \int \frac{1}{4 (1 + u^{2})} d u$

Step 6

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$6 (\frac{1}{4} \int \frac{1}{1 + u^{2}} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{4}$ and $6$ .

$\frac{6}{4} \int \frac{1}{1 + u^{2}} d u$

Step 7.1.2

Cancel the common factor of $6$ and $4$ .

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Step 7.1.2.1

Factor $2$ out of $6$ .

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

Step 7.1.2.2

Cancel the common factors.

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Step 7.1.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 3}{2 \cdot 2} \int \frac{1}{1 + u^{2}} d u$

Step 7.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 2} \int \frac{1}{1 + u^{2}} d u$

Step 7.1.2.2.3

Rewrite the expression.

$\frac{3}{2} \int \frac{1}{1 + u^{2}} d u$

Step 7.2

Rewrite $1$ as $1^{2}$ .

$\frac{3}{2} \int \frac{1}{1^{2} + u^{2}} d u$

Step 8

The integral of $\frac{1}{1^{2} + u^{2}}$ with respect to $u$ is $\arctan (u) + C$ .

$\frac{3}{2} (\arctan (u) + C)$

Step 9

Simplify.

$\frac{3}{2} \arctan (u) + C$

Step 10

Replace all occurrences of $u$ with $x^{4}$ .

$\frac{3}{2} \arctan (x^{4}) + C$