Calculus Examples

$\frac{1}{x^{2} + 4}$

Step 1

Write $\frac{1}{x^{2} + 4}$ as a function.

$f (x) = \frac{1}{x^{2} + 4}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int \frac{1}{x^{2} + 4} d x$

Step 4

Simplify the expression.

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Step 4.1

Reorder $x^{2}$ and $4$ .

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

Step 4.2

Rewrite $4$ as $2^{2}$ .

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

Step 5

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

$\frac{1}{2} \arctan (\frac{x}{2}) + C$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{2}$ and $\arctan (\frac{x}{2})$ .

$\frac{\arctan (\frac{x}{2})}{2} + C$

Step 6.2

Rewrite $\frac{\arctan (\frac{x}{2})}{2} + C$ as $\frac{1}{2} \arctan (\frac{1}{2} x) + C$ .

$\frac{1}{2} \arctan (\frac{1}{2} x) + C$

Step 7

The answer is the antiderivative of the function $f (x) = \frac{1}{x^{2} + 4}$ .

$F (x) =$ $\frac{1}{2} \arctan (\frac{1}{2} x) + C$