Calculus Examples

$\int 8 \tan^{3} (x) d x$

Step 1

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

$8 \int \tan^{3} (x) d x$

Step 2

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

$8 \int \tan^{2} (x) \tan (x) d x$

Step 3

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

$8 \int (- 1 + \sec^{2} (x)) \tan (x) d x$

Step 4

Apply the distributive property.

$8 \int - 1 \tan (x) + \sec^{2} (x) \tan (x) d x$

Step 5

Split the single integral into multiple integrals.

$8 (\int - 1 \tan (x) d x + \int \sec^{2} (x) \tan (x) d x)$

Step 6

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

$8 (- \int \tan (x) d x + \int \sec^{2} (x) \tan (x) d x)$

Step 7

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$8 (- (\ln (| \sec (x) |) + C) + \int \sec^{2} (x) \tan (x) d x)$

Step 8

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

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Step 8.1

Let $u = \sec (x)$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $\sec (x)$ .

$\frac{d}{d x} [\sec (x)]$

Step 8.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 8.2

Rewrite the problem using $u$ and $d u$ .

$8 (- (\ln (| \sec (x) |) + C) + \int u d u)$

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $u^{2}$ .

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

Step 10.2

Simplify.

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

Step 11

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Multiply $- 1$ by $8$ .

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

Step 12.4

Cancel the common factor of $2$ .

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Step 12.4.1

Factor $2$ out of $8$ .

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

Step 12.4.2

Cancel the common factor.

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

Step 12.4.3

Rewrite the expression.

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