Calculus Examples

$\int x \arctan ({(x)}^{2}) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2}$ .

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

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = \arctan (u_{1})$ and $dv = 1$ .

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

Step 5

Combine $u_{1}$ and $\frac{1}{{u_{1}}^{2} + 1}$ .

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate ${u_{1}}^{2} + 1$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2} + 1]$

Step 6.1.2

By the Sum Rule, the derivative of ${u_{1}}^{2} + 1$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [{u_{1}}^{2}] + \frac{d}{d u_{1}} [1]$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] + \frac{d}{d u_{1}} [1]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} + \frac{d}{d u_{1}} [1]$

Step 6.1.4

Since $1$ is constant with respect to $u_{1}$ , the derivative of $1$ with respect to $u_{1}$ is $0$ .

$2 u_{1} + 0$

Step 6.1.5

Add $2 u_{1}$ and $0$ .

$2 u_{1}$

Step 6.2

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

$\frac{1}{2} (\arctan (u_{1}) u_{1} - \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2})$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{2} (\arctan (u_{1}) u_{1} - \int \frac{1}{u_{2} \cdot 2} d u_{2})$

Step 7.2

Move $2$ to the left of $u_{2}$ .

$\frac{1}{2} (\arctan (u_{1}) u_{1} - \int \frac{1}{2 u_{2}} d u_{2})$

Step 8

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

$\frac{1}{2} (\arctan (u_{1}) u_{1} - (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2}))$

Step 9

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{2} (\arctan (u_{1}) u_{1} - \frac{1}{2} (\ln (| u_{2} |) + C))$

Step 10

Simplify.

$\frac{1}{2} (\arctan (u_{1}) u_{1} - \frac{1}{2} \ln (| u_{2} |)) + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

$\frac{1}{2} (\arctan (x^{2}) x^{2} - \frac{1}{2} \ln (| u_{2} |)) + C$

Step 11.2

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

$\frac{1}{2} (\arctan (x^{2}) x^{2} - \frac{1}{2} \ln (| {u_{1}}^{2} + 1 |)) + C$

Step 11.3

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

$\frac{1}{2} (\arctan (x^{2}) x^{2} - \frac{1}{2} \ln (| {(x^{2})}^{2} + 1 |)) + C$

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 12.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} (\arctan (x^{2}) x^{2} - \frac{1}{2} \ln (| x^{2 \cdot 2} + 1 |)) + C$

Step 12.1.1.2

Multiply $2$ by $2$ .

$\frac{1}{2} (\arctan (x^{2}) x^{2} - \frac{1}{2} \ln (| x^{4} + 1 |)) + C$

Step 12.1.2

Combine $\ln (| x^{4} + 1 |)$ and $\frac{1}{2}$ .

$\frac{1}{2} (\arctan (x^{2}) x^{2} - \frac{\ln (| x^{4} + 1 |)}{2}) + C$

Step 12.2

To write $\arctan (x^{2}) x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (\arctan (x^{2}) x^{2} \cdot \frac{2}{2} - \frac{\ln (| x^{4} + 1 |)}{2}) + C$

Step 12.3

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

$\frac{1}{2} (\frac{\arctan (x^{2}) x^{2} \cdot 2}{2} - \frac{\ln (| x^{4} + 1 |)}{2}) + C$

Step 12.4

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{\arctan (x^{2}) x^{2} \cdot 2 - \ln (| x^{4} + 1 |)}{2} + C$

Step 12.5

Move $2$ to the left of $\arctan (x^{2}) x^{2}$ .

$\frac{1}{2} \cdot \frac{2 \arctan (x^{2}) x^{2} - \ln (| x^{4} + 1 |)}{2} + C$

Step 12.6

Multiply $\frac{1}{2} \cdot \frac{2 \arctan (x^{2}) x^{2} - \ln (| x^{4} + 1 |)}{2}$ .

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Step 12.6.1

Multiply $\frac{1}{2}$ by $\frac{2 \arctan (x^{2}) x^{2} - \ln (| x^{4} + 1 |)}{2}$ .

$\frac{2 \arctan (x^{2}) x^{2} - \ln (| x^{4} + 1 |)}{2 \cdot 2} + C$

Step 12.6.2

Multiply $2$ by $2$ .

$\frac{2 \arctan (x^{2}) x^{2} - \ln (| x^{4} + 1 |)}{4} + C$

Step 12.7

Reorder factors in $\frac{2 \arctan (x^{2}) x^{2} - \ln (| x^{4} + 1 |)}{4}$ .

$\frac{2 x^{2} \arctan (x^{2}) - \ln (| x^{4} + 1 |)}{4} + C$

Step 13

Reorder terms.

$\frac{1}{4} (2 x^{2} \arctan (x^{2}) - \ln (| x^{4} + 1 |)) + C$