Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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 4

Simplify the answer.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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 4.3

Reorder terms.

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