Calculus Examples

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

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

Step 1

Factor out

{tan}^{2} (x)

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

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

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

Step 2

Using the Pythagorean Identity, rewrite

{tan}^{2} (x)

$\tan^{2} (x)$ as

- 1 + {sec}^{2} (x)

$- 1 + \sec^{2} (x)$ .

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

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

Step 3

Apply the distributive property.

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

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

Step 4

Split the single integral into multiple integrals.

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

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

Step 5

Since

- 1

$- 1$ is constant with respect to

x

$x$ , move

- 1

$- 1$ out of the integral.

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

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

Step 6

The integral of

tan (x)

$\tan (x)$ with respect to

x

$x$ is

ln (| sec (x) |)

$\ln (| \sec (x) |)$ .

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

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

Step 7

Let

u = sec (x)

$u = \sec (x)$ . Then

d u = sec (x) tan (x) d x

$d u = \sec (x) \tan (x) d x$ , so

\frac{1}{sec (x) tan (x)} d u = d x

$\frac{1}{\sec (x) \tan (x)} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 7.1

Let

u = sec (x)

$u = \sec (x)$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 7.1.1

Differentiate

sec (x)

$\sec (x)$ .

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

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

Step 7.1.2

The derivative of

sec (x)

$\sec (x)$ with respect to

x

$x$ is

sec (x) tan (x)

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

sec (x) tan (x)

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

sec (x) tan (x)

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

Step 7.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

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

Step 8

By the Power Rule, the integral of

u

$u$ with respect to

u

$u$ is

\frac{1}{2} u^{2}

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

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

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

Step 9

Simplify.

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

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

Step 10

Replace all occurrences of

u

$u$ with

sec (x)

$\sec (x)$ .

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

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