Calculus Examples

$\int \frac{8 \arctan (2 x)}{1 + 4 x^{2}} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int \frac{\arctan (2 x)}{1 + 4 x^{2}} d x$

Step 2

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

$8 \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} \cdot \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}}$ by $\frac{1}{2}$ .

$8 \int \frac{\arctan (u_{1})}{(1 + 4 {(\frac{u_{1}}{2})}^{2}) \cdot 2} d u_{1}$

Step 3.2

Move $2$ to the left of $1 + 4 {(\frac{u_{1}}{2})}^{2}$ .

$8 \int \frac{\arctan (u_{1})}{2 (1 + 4 {(\frac{u_{1}}{2})}^{2})} d u_{1}$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$8 (\frac{1}{2} \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} d u_{1})$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $8$ .

$\frac{8}{2} \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} d u_{1}$

Step 5.2

Cancel the common factor of $8$ and $2$ .

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Step 5.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2} \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} d u_{1}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 4}{2 (1)} \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} d u_{1}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 1} \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} d u_{1}$

Step 5.2.2.3

Rewrite the expression.

$\frac{4}{1} \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} d u_{1}$

Step 5.2.2.4

Divide $4$ by $1$ .

$4 \int \frac{\arctan (u_{1})}{1 + 4 {(\frac{u_{1}}{2})}^{2}} d u_{1}$

Step 6

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 $d$ $u_{2}$ .

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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 6.1.3

Reorder terms.

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

Step 6.1.4

Rearrange terms.

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

Step 6.2

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

$4 \int u_{2} d u_{2}$

Step 7

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

$4 (\frac{1}{2} {u_{2}}^{2} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $4 (\frac{1}{2} {u_{2}}^{2} + C)$ as $4 (\frac{1}{2}) {u_{2}}^{2} + C$ .

$4 (\frac{1}{2}) {u_{2}}^{2} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} {u_{2}}^{2} + C$

Step 8.2.2

Cancel the common factor of $4$ and $2$ .

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Step 8.2.2.1

Factor $2$ out of $4$ .

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

Step 8.2.2.2

Cancel the common factors.

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Step 8.2.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.2.2.2

Cancel the common factor.

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

Step 8.2.2.2.3

Rewrite the expression.

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

Step 8.2.2.2.4

Divide $2$ by $1$ .

$2 {u_{2}}^{2} + C$

Step 9

Substitute back in for each integration substitution variable.

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Step 9.1

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

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

Step 9.2

Replace all occurrences of $u_{1}$ with $2 x$ .

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