Calculus Examples

$\int_{0}^{\frac{π}{2}} \sin^{3} (x) d x$

Step 1

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

$\int_{0}^{\frac{π}{2}} \sin^{2} (x) \sin (x) d x$

Step 2

Using the Pythagorean Identity, rewrite $\sin^{2} (x)$ as $1 - \cos^{2} (x)$ .

$\int_{0}^{\frac{π}{2}} (1 - \cos^{2} (x)) \sin (x) d x$

Step 3

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Differentiate $\cos (x)$ .

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

Step 3.1.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- \sin (x)$

Step 3.2

Substitute the lower limit in for $x$ in $u = \cos (x)$ .

$u_{lower} = \cos (0)$

Step 3.3

The exact value of $\cos (0)$ is $1$ .

$u_{lower} = 1$

Step 3.4

Substitute the upper limit in for $x$ in $u = \cos (x)$ .

$u_{upper} = \cos (\frac{π}{2})$

Step 3.5

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$u_{upper} = 0$

Step 3.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 0$

Step 3.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{0} - 1 + u^{2} d u$

Step 4

Split the single integral into multiple integrals.

$\int_{1}^{0} - 1 d u + \int_{1}^{0} u^{2} d u$

Step 5

Apply the constant rule.

$- u]_{1}^{0} + \int_{1}^{0} u^{2} d u$

Step 6

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$- u]_{1}^{0} + \frac{1}{3} u^{3}]_{1}^{0}$

Step 7

Combine $- u]_{1}^{0}$ and $\frac{1}{3} u^{3}]_{1}^{0}$ .

$- u + \frac{1}{3} u^{3}]_{1}^{0}$

Step 8

Substitute and simplify.

Tap for more steps...

Step 8.1

Evaluate $- u + \frac{1}{3} u^{3}$ at $0$ and at $1$ .

$(- 0 + \frac{1}{3} \cdot 0^{3}) - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3})$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Multiply $- 1$ by $0$ .

$0 + \frac{1}{3} \cdot 0^{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3})$

Step 8.2.2

Raising $0$ to any positive power yields $0$ .

$0 + \frac{1}{3} \cdot 0 - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3})$

Step 8.2.3

Multiply $\frac{1}{3}$ by $0$ .

$0 + 0 - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3})$

Step 8.2.4

Add $0$ and $0$ .

$0 - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3})$

Step 8.2.5

Multiply $- 1$ by $1$ .

$0 - (- 1 + \frac{1}{3} \cdot 1^{3})$

Step 8.2.6

One to any power is one.

$0 - (- 1 + \frac{1}{3} \cdot 1)$

Step 8.2.7

Multiply $\frac{1}{3}$ by $1$ .

$0 - (- 1 + \frac{1}{3})$

Step 8.2.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$0 - (- 1 \cdot \frac{3}{3} + \frac{1}{3})$

Step 8.2.9

Combine $- 1$ and $\frac{3}{3}$ .

$0 - (\frac{- 1 \cdot 3}{3} + \frac{1}{3})$

Step 8.2.10

Combine the numerators over the common denominator.

$0 - \frac{- 1 \cdot 3 + 1}{3}$

Step 8.2.11

Simplify the numerator.

Tap for more steps...

Step 8.2.11.1

Multiply $- 1$ by $3$ .

$0 - \frac{- 3 + 1}{3}$

Step 8.2.11.2

Add $- 3$ and $1$ .

$0 - \frac{- 2}{3}$

Step 8.2.12

Move the negative in front of the fraction.

$0 - - \frac{2}{3}$

Step 8.2.13

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{2}{3})$

Step 8.2.14

Multiply $\frac{2}{3}$ by $1$ .

$0 + \frac{2}{3}$

Step 8.2.15

Add $0$ and $\frac{2}{3}$ .

$\frac{2}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$