Calculus Examples

$\int \frac{1}{x^{2} + 5} d x$

Step 1

Reorder $x^{2}$ and $5$ .

$\int \frac{1}{5 + x^{2}} d x$

Step 2

Rewrite $5$ as ${\sqrt{5}}^{2}$ .

$\int \frac{1}{{\sqrt{5}}^{2} + x^{2}} d x$

Step 3

The integral of $\frac{1}{{\sqrt{5}}^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{\sqrt{5}} \arctan (\frac{x}{\sqrt{5}}) + C$ .

$\frac{1}{\sqrt{5}} \arctan (\frac{x}{\sqrt{5}}) + C$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Combine $\frac{1}{\sqrt{5}}$ and $\arctan (\frac{x}{\sqrt{5}})$ .

$\frac{\arctan (\frac{x}{\sqrt{5}})}{\sqrt{5}} + C$

Step 4.2

Rewrite $\frac{\arctan (\frac{x}{\sqrt{5}})}{\sqrt{5}} + C$ as $\frac{\arctan (\frac{x \sqrt{5}}{5})}{\sqrt{5}} + C$ .

$\frac{\arctan (\frac{x \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 4.3

Reorder terms.

$\frac{1}{\sqrt{5}} \arctan (\frac{\sqrt{5}}{5} x) + C$