Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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

Step 5

Rewrite $2$ as ${\sqrt{2}}^{2}$ .

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Reorder terms.

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

Step 8

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

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