Calculus Examples

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

Step 1

Split the fraction $\frac{2 x + 4}{x^{2} + 9}$ into two fractions.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $x^{2} + 9$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 x + 0$

Step 4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 4.2

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

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

Step 8

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

$\ln (| u |) + C + \int \frac{4}{x^{2} + 9} d x$

Step 9

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

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

Step 10

Simplify the expression.

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Step 10.1

Reorder $x^{2}$ and $9$ .

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

Step 10.2

Rewrite $9$ as $3^{2}$ .

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

Step 11

The integral of $\frac{1}{3^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{3} \arctan (\frac{x}{3}) + C$ .

$\ln (| u |) + C + 4 (\frac{1}{3} \arctan (\frac{x}{3}) + C)$

Step 12

Simplify.

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Step 12.1

Combine $\frac{1}{3}$ and $\arctan (\frac{x}{3})$ .

$\ln (| u |) + C + 4 (\frac{\arctan (\frac{x}{3})}{3} + C)$

Step 12.2

Simplify.

$\ln (| u |) + \frac{4 \arctan (\frac{x}{3})}{3} + C$

Step 13

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

$\ln (| x^{2} + 9 |) + \frac{4 \arctan (\frac{x}{3})}{3} + C$

Step 14

Reorder terms.

$\ln (| x^{2} + 9 |) + \frac{4}{3} \arctan (\frac{1}{3} x) + C$