Calculus Examples

$\int \tan^{3} (4 x) d x$

Step 1

Let $u_{1} = 4 x$ . Then $d u_{1} = 4 d x$ , so $\frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 4 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 1.1.3

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

$4 \cdot 1$

Step 1.1.4

Multiply $4$ by $1$ .

$4$

Step 1.2

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

$\int \tan^{3} (u_{1}) \frac{1}{4} d u_{1}$

Step 2

Combine $\tan^{3} (u_{1})$ and $\frac{1}{4}$ .

$\int \frac{\tan^{3} (u_{1})}{4} d u_{1}$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \tan^{3} (u_{1}) d u_{1}$

Step 4

Factor out $\tan^{2} (u_{1})$ .

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

Step 5

Using the Pythagorean Identity, rewrite $\tan^{2} (u_{1})$ as $- 1 + \sec^{2} (u_{1})$ .

$\frac{1}{4} \int (- 1 + \sec^{2} (u_{1})) \tan (u_{1}) d u_{1}$

Step 6

Apply the distributive property.

$\frac{1}{4} \int - 1 \tan (u_{1}) + \sec^{2} (u_{1}) \tan (u_{1}) d u_{1}$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{4} (\int - 1 \tan (u_{1}) d u_{1} + \int \sec^{2} (u_{1}) \tan (u_{1}) d u_{1})$

Step 8

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

$\frac{1}{4} (- \int \tan (u_{1}) d u_{1} + \int \sec^{2} (u_{1}) \tan (u_{1}) d u_{1})$

Step 9

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

$\frac{1}{4} (- (\ln (| \sec (u_{1}) |) + C) + \int \sec^{2} (u_{1}) \tan (u_{1}) d u_{1})$

Step 10

Let $u_{2} = \sec (u_{1})$ . Then $d u_{2} = \sec (u_{1}) \tan (u_{1}) d u_{1}$ , so $\frac{1}{\sec (u_{1}) \tan (u_{1})} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$\sec (u_{1}) \tan (u_{1})$

Step 10.2

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

$\frac{1}{4} (- (\ln (| \sec (u_{1}) |) + C) + \int u_{2} d u_{2})$

Step 11

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

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

Step 12

Simplify.

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

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{1}$ with $4 x$ .

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

Step 13.2

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

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

Step 13.3

Replace all occurrences of $u_{1}$ with $4 x$ .

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

Step 14

Simplify.

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Step 14.1

Combine $\frac{1}{2}$ and $\sec^{2} (4 x)$ .

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

Step 14.2

Apply the distributive property.

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

Step 14.3

Combine $\frac{1}{4}$ and $\ln (| \sec (4 x) |)$ .

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

Step 14.4

Combine.

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

Step 14.5

Simplify each term.

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Step 14.5.1

Multiply $\sec^{2} (4 x)$ by $1$ .

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

Step 14.5.2

Multiply $4$ by $2$ .

$- \frac{\ln (| \sec (4 x) |)}{4} + \frac{\sec^{2} (4 x)}{8} + C$

Step 15

Reorder terms.

$- \frac{1}{4} \ln (| \sec (4 x) |) + \frac{1}{8} \sec^{2} (4 x) + C$