Calculus Examples

$\int \frac{\arctan (3 x)}{1 + 9 x^{2}} d x$

Step 1

Let $u_{1} = 3 x$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 1.1

Let $u_{1} = 3 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 1.1.4

Multiply $3$ by $1$ .

$3$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{\arctan (u_{1})}{1 + 9 {(\frac{u_{1}}{3})}^{2}} \cdot \frac{1}{3} d u_{1}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply $\frac{\arctan (u_{1})}{1 + 9 {(\frac{u_{1}}{3})}^{2}}$ by $\frac{1}{3}$ .

$\int \frac{\arctan (u_{1})}{(1 + 9 {(\frac{u_{1}}{3})}^{2}) \cdot 3} d u_{1}$

Step 2.2

Move $3$ to the left of $1 + 9 {(\frac{u_{1}}{3})}^{2}$ .

$\int \frac{\arctan (u_{1})}{3 (1 + 9 {(\frac{u_{1}}{3})}^{2})} d u_{1}$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{\arctan (u_{1})}{1 + 9 {(\frac{u_{1}}{3})}^{2}} d u_{1}$

Step 4

Let $u_{2} = \arctan (u_{1})$ . Then $d u_{2} = \frac{1}{1 + {u_{1}}^{2}} d u_{1}$ , so $- d u_{2} = \frac{- 1}{{u_{1}}^{2} + 1} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 4.1

Let $u_{2} = \arctan (u_{1})$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 4.1.1

Differentiate $\arctan (u_{1})$ .

$\frac{d}{d u_{1}} [\arctan (u_{1})]$

Step 4.1.2

The derivative of $\arctan (u_{1})$ with respect to $u_{1}$ is $\frac{1}{1 + {u_{1}}^{2}}$ .

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

Step 4.1.3

Reorder terms.

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

Step 4.1.4

Rearrange terms.

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

Step 4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{3} \int u_{2} d u_{2}$

Step 5

By the Power Rule, the integral of $u_{2}$ with respect to $u_{2}$ is $\frac{1}{2} {u_{2}}^{2}$ .

$\frac{1}{3} (\frac{1}{2} {u_{2}}^{2} + C)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Rewrite $\frac{1}{3} (\frac{1}{2} {u_{2}}^{2} + C)$ as $\frac{1}{3} \cdot \frac{1}{2} {u_{2}}^{2} + C$ .

$\frac{1}{3} \cdot \frac{1}{2} {u_{2}}^{2} + C$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$\frac{1}{3 \cdot 2} {u_{2}}^{2} + C$

Step 6.2.2

Multiply $3$ by $2$ .

$\frac{1}{6} {u_{2}}^{2} + C$

Step 7

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 7.1

Replace all occurrences of $u_{2}$ with $\arctan (u_{1})$ .

$\frac{1}{6} {\arctan (u_{1})}^{2} + C$

Step 7.2

Replace all occurrences of $u_{1}$ with $3 x$ .

$\frac{1}{6} {\arctan (3 x)}^{2} + C$