Calculus Examples

$\int \frac{9 x}{{(x^{2} + 1)}^{4}} d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int \frac{x}{{(x^{2} + 1)}^{4}} d x$

Step 2

Let $u = x^{2} + 1$ . 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 2.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 2.1.2

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

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 2.1.3

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

$2 x + \frac{d}{d x} [1]$

Step 2.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.2

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

$9 \int \frac{1}{u^{4}} \cdot \frac{1}{2} d u$

Step 3

Simplify.

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Step 3.1

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

$9 \int \frac{1}{u^{4} \cdot 2} d u$

Step 3.2

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

$9 \int \frac{1}{2 u^{4}} d u$

Step 4

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

$9 (\frac{1}{2} \int \frac{1}{u^{4}} d u)$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} \int \frac{1}{u^{4}} d u$

Step 5.2

Apply basic rules of exponents.

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Step 5.2.1

Move $u^{4}$ out of the denominator by raising it to the $- 1$ power.

$\frac{9}{2} \int {(u^{4})}^{- 1} d u$

Step 5.2.2

Multiply the exponents in ${(u^{4})}^{- 1}$ .

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Step 5.2.2.1

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

$\frac{9}{2} \int u^{4 \cdot - 1} d u$

Step 5.2.2.2

Multiply $4$ by $- 1$ .

$\frac{9}{2} \int u^{- 4} d u$

Step 6

By the Power Rule, the integral of $u^{- 4}$ with respect to $u$ is $- \frac{1}{3} u^{- 3}$ .

$\frac{9}{2} (- \frac{1}{3} u^{- 3} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{9}{2} (- \frac{1}{3} u^{- 3} + C)$ as $\frac{9}{2} (- \frac{1}{3} \cdot \frac{1}{u^{3}}) + C$ .

$\frac{9}{2} (- \frac{1}{3} \cdot \frac{1}{u^{3}}) + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{1}{u^{3}}$ by $\frac{1}{3}$ .

$\frac{9}{2} (- \frac{1}{u^{3} \cdot 3}) + C$

Step 7.2.2

Move $3$ to the left of $u^{3}$ .

$\frac{9}{2} (- \frac{1}{3 \cdot u^{3}}) + C$

Step 7.2.3

Multiply $\frac{9}{2}$ by $\frac{1}{3 u^{3}}$ .

$- \frac{9}{2 (3 u^{3})} + C$

Step 7.2.4

Multiply $3$ by $2$ .

$- \frac{9}{6 u^{3}} + C$

Step 7.2.5

Cancel the common factor of $9$ and $6$ .

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Step 7.2.5.1

Factor $3$ out of $9$ .

$- \frac{3 (3)}{6 u^{3}} + C$

Step 7.2.5.2

Cancel the common factors.

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Step 7.2.5.2.1

Factor $3$ out of $6 u^{3}$ .

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

Step 7.2.5.2.2

Cancel the common factor.

$- \frac{3 \cdot 3}{3 (2 u^{3})} + C$

Step 7.2.5.2.3

Rewrite the expression.

$- \frac{3}{2 u^{3}} + C$

Step 8

Replace all occurrences of $u$ with $x^{2} + 1$ .

$- \frac{3}{2 {(x^{2} + 1)}^{3}} + C$