Calculus Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

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

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