Calculus Examples

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

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2

Let $u = x^{2} + 9$ . 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 $u = x^{2} + 9$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{2} + 9$ .

$\frac{d}{d x} [x^{2} + 9]$

Step 2.1.2

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

Step 2.1.3

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

$2 x + \frac{d}{d x} [9]$

Step 2.1.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ 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 $d u$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Move $2$ to the left of $u$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $4$ and $2$ .

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Step 5.2.1

Factor $2$ out of $4$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} \int \frac{1}{u} d u$

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.2.4

Divide $2$ by $1$ .

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

Step 6

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

$2 (\ln (| u |) + C)$

Step 7

Simplify.

$2 \ln (| u |) + C$

Step 8

Replace all occurrences of $u$ with $x^{2} + 9$ .

$2 \ln (| x^{2} + 9 |) + C$