Calculus Examples

$\int \frac{3}{x^{2} + 9} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{1}{x^{2} + 9} d x$

Step 2

Simplify the expression.

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Step 2.1

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

$3 \int \frac{1}{9 + x^{2}} d x$

Step 2.2

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

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

Step 3

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

$3 (\frac{1}{3} \arctan (\frac{x}{3}) + C)$

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Rewrite $3 (\frac{\arctan (\frac{x}{3})}{3} + C)$ as $3 (\frac{1}{3}) \arctan (\frac{1}{3} x) + C$ .

$3 (\frac{1}{3}) \arctan (\frac{1}{3} x) + C$

Step 4.3

Simplify.

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Step 4.3.1

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} \arctan (\frac{1}{3} x) + C$

Step 4.3.2

Cancel the common factor of $3$ .

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Step 4.3.2.1

Cancel the common factor.

$\frac{3}{3} \arctan (\frac{1}{3} x) + C$

Step 4.3.2.2

Rewrite the expression.

$1 \arctan (\frac{1}{3} x) + C$

Step 4.3.3

Combine $\frac{1}{3}$ and $x$ .

$1 \arctan (\frac{x}{3}) + C$

Step 4.3.4

Multiply $\arctan (\frac{x}{3})$ by $1$ .

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

Step 4.4

Reorder terms.

$\arctan (\frac{1}{3} x) + C$