Calculus Examples

$f (x) = \frac{1}{x^{2} + 36}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int \frac{1}{x^{2} + 36} d x$

Step 3

Simplify the expression.

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Step 3.1

Reorder $x^{2}$ and $36$ .

$\int \frac{1}{36 + x^{2}} d x$

Step 3.2

Rewrite $36$ as $6^{2}$ .

$\int \frac{1}{6^{2} + x^{2}} d x$

Step 4

The integral of $\frac{1}{6^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{6} \arctan (\frac{x}{6}) + C$ .

$\frac{1}{6} \arctan (\frac{x}{6}) + C$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{6}$ and $\arctan (\frac{x}{6})$ .

$\frac{\arctan (\frac{x}{6})}{6} + C$

Step 5.2

Rewrite $\frac{\arctan (\frac{x}{6})}{6} + C$ as $\frac{1}{6} \arctan (\frac{1}{6} x) + C$ .

$\frac{1}{6} \arctan (\frac{1}{6} x) + C$

Step 6

The answer is the antiderivative of the function $f (x) = \frac{1}{x^{2} + 36}$ .

$F (x) =$ $\frac{1}{6} \arctan (\frac{1}{6} x) + C$