Calculus Examples

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

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

Step 1

Factor out

\sin^{2} (x)

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

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

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

Step 2

Using the Pythagorean Identity, rewrite

\sin^{2} (x)

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

1 - \cos^{2} (x)

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

\int (1 - \cos^{2} (x)) \sin (x) d x

$\int (1 - \cos^{2} (x)) \sin (x) d x$

Step 3

Let

u = \cos (x)

$u = \cos (x)$ . Then

d u = - \sin (x) d x

$d u = - \sin (x) d x$ , so

- \frac{1}{\sin (x)} d u = d x

$- \frac{1}{\sin (x)} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

Tap for more steps...

Step 3.1

Let

u = \cos (x)

$u = \cos (x)$ . Find

\frac{d u}{d x}

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

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Step 3.1.1

Differentiate

\cos (x)

$\cos (x)$ .

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

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

Step 3.1.2

The derivative of

\cos (x)

$\cos (x)$ with respect to

x

$x$ is

- \sin (x)

$- \sin (x)$ .

- \sin (x)

$- \sin (x)$

- \sin (x)

$- \sin (x)$

Step 3.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\int - 1 + u^{2} d u

$\int - 1 + u^{2} d u$

\int - 1 + u^{2} d u

$\int - 1 + u^{2} d u$

Step 4

Split the single integral into multiple integrals.

\int - 1 d u + \int u^{2} d u

$\int - 1 d u + \int u^{2} d u$

Step 5

Apply the constant rule.

- u + C + \int u^{2} d u

$- u + C + \int u^{2} d u$

Step 6

By the Power Rule, the integral of

u^{2}

$u^{2}$ with respect to

u

$u$ is

\frac{1}{3} u^{3}

$\frac{1}{3} u^{3}$ .

- u + C + \frac{1}{3} u^{3} + C

$- u + C + \frac{1}{3} u^{3} + C$

Step 7

Simplify.

- u + \frac{1}{3} u^{3} + C

$- u + \frac{1}{3} u^{3} + C$

Step 8

Replace all occurrences of

u

$u$ with

\cos (x)

$\cos (x)$ .

- \cos (x) + \frac{1}{3} \cos^{3} (x) + C

$- \cos (x) + \frac{1}{3} \cos^{3} (x) + C$

$\int_{}^{} \sin^{3} x d x$

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[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$