Calculus Examples

$\int_{0}^{\frac{π}{2}} \cos^{11} (x) \sin^{5} (x) d x$

Step 1

Factor out

$\cos^{10} (x)$ .

$\int_{0}^{\frac{π}{2}} \cos^{10} (x) \cos (x) \sin^{5} (x) d x$

Step 2

Simplify with factoring out.

Tap for more steps...

Step 2.1

Factor

$2$ out of

$10$ .

$\int_{0}^{\frac{π}{2}} {\cos (x)}^{2 (5)} \cos (x) \sin^{5} (x) d x$

Step 2.2

Rewrite

${\cos (x)}^{2 (5)}$ as exponentiation.

$\int_{0}^{\frac{π}{2}} {(\cos^{2} (x))}^{5} \cos (x) \sin^{5} (x) d x$

Step 3

Using the Pythagorean Identity, rewrite

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

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

$\int_{0}^{\frac{π}{2}} {(1 - \sin^{2} (x))}^{5} \cos (x) \sin^{5} (x) d x$

Step 4

Let

$u_{1} = \sin (x)$ . Then

$d u_{1} = \cos (x) d x$ , so

$\frac{1}{\cos (x)} d u_{1} = d x$ . Rewrite using

$u_{1}$ and

$d$

$u_{1}$ .

Tap for more steps...

Step 4.1

Let

$u_{1} = \sin (x)$ . Find

$\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate

$\sin (x)$ .

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

Step 4.1.2

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

$\cos (x)$

Step 4.2

Substitute the lower limit in for

$x$ in

$u_{1} = \sin (x)$ .

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

Step 4.3

The exact value of

$\sin (0)$ is

$0$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for

$x$ in

$u_{1} = \sin (x)$ .

$u_{upper} = \sin (\frac{π}{2})$

Step 4.5

The exact value of

$\sin (\frac{π}{2})$ is

$1$ .

$u_{upper} = 1$

Step 4.6

The values found for

$u_{lower}$ and

$u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 4.7

Rewrite the problem using

$u_{1}$ ,

$d u_{1}$ , and the new limits of integration.

$\int_{0}^{1} {(1 - {u_{1}}^{2})}^{5} {u_{1}}^{5} d u_{1}$

Step 5

Let

$u_{2} = 1 - {u_{1}}^{2}$ . Then

$d u_{2} = - 2 u_{1} d u_{1}$ , so

$- \frac{1}{2} d u_{2} = u_{1} d u_{1}$ . Rewrite using

$u_{2}$ and

$d$

$u_{2}$ .

Tap for more steps...

Step 5.1

Let

$u_{2} = 1 - {u_{1}}^{2}$ . Find

$\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 5.1.1

Differentiate

$1 - {u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [1 - {u_{1}}^{2}]$

Step 5.1.2

Differentiate.

Tap for more steps...

Step 5.1.2.1

By the Sum Rule, the derivative of

$1 - {u_{1}}^{2}$ with respect to

$u_{1}$ is

$\frac{d}{d u_{1}} [1] + \frac{d}{d u_{1}} [- {u_{1}}^{2}]$ .

$\frac{d}{d u_{1}} [1] + \frac{d}{d u_{1}} [- {u_{1}}^{2}]$

Step 5.1.2.2

Since

$1$ is constant with respect to

$u_{1}$ , the derivative of

$1$ with respect to

$u_{1}$ is

$0$ .

$0 + \frac{d}{d u_{1}} [- {u_{1}}^{2}]$

Step 5.1.3

Evaluate

$\frac{d}{d u_{1}} [- {u_{1}}^{2}]$ .

Tap for more steps...

Step 5.1.3.1

Since

$- 1$ is constant with respect to

$u_{1}$ , the derivative of

$- {u_{1}}^{2}$ with respect to

$u_{1}$ is

$- \frac{d}{d u_{1}} [{u_{1}}^{2}]$ .

$0 - \frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 5.1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is

$n {u_{1}}^{n - 1}$ where

$n = 2$ .

$0 - (2 u_{1})$

Step 5.1.3.3

Multiply

$2$ by

$- 1$ .

$0 - 2 u_{1}$

Step 5.1.4

Subtract

$2 u_{1}$ from

$0$ .

$- 2 u_{1}$

Step 5.2

Substitute the lower limit in for

$u_{1}$ in

$u_{2} = 1 - {u_{1}}^{2}$ .

$u_{lower} = 1 - 0^{2}$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

Raising

$0$ to any positive power yields

$0$ .

$u_{lower} = 1 - 0$

Step 5.3.1.2

Multiply

$- 1$ by

$0$ .

$u_{lower} = 1 + 0$

Step 5.3.2

Add

$1$ and

$0$ .

$u_{lower} = 1$

Step 5.4

Substitute the upper limit in for

$u_{1}$ in

$u_{2} = 1 - {u_{1}}^{2}$ .

$u_{upper} = 1 - 1^{2}$

Step 5.5

Simplify.

Tap for more steps...

Step 5.5.1

Simplify each term.

Tap for more steps...

Step 5.5.1.1

One to any power is one.

$u_{upper} = 1 - 1 \cdot 1$

Step 5.5.1.2

Multiply

$- 1$ by

$1$ .

$u_{upper} = 1 - 1$

Step 5.5.2

Subtract

$1$ from

$1$ .

$u_{upper} = 0$

Step 5.6

The values found for

$u_{lower}$ and

$u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 0$

Step 5.7

Rewrite the problem using

$u_{2}$ ,

$d u_{2}$ , and the new limits of integration.

$\int_{1}^{0} {u_{2}}^{5} {\sqrt{- u_{2} + 1}}^{4} \frac{1}{- 2} d u_{2}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Rewrite

${\sqrt{- u_{2} + 1}}^{4}$ as

${(- u_{2} + 1)}^{2}$ .

Tap for more steps...

Step 6.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{- u_{2} + 1}$ as

${(- u_{2} + 1)}^{\frac{1}{2}}$ .

$\int_{1}^{0} {u_{2}}^{5} {({(- u_{2} + 1)}^{\frac{1}{2}})}^{4} \frac{1}{- 2} d u_{2}$

Step 6.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{\frac{1}{2} \cdot 4} \frac{1}{- 2} d u_{2}$

Step 6.1.3

Combine

$\frac{1}{2}$ and

$4$ .

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{\frac{4}{2}} \frac{1}{- 2} d u_{2}$

Step 6.1.4

Cancel the common factor of

$4$ and

$2$ .

Tap for more steps...

Step 6.1.4.1

Factor

$2$ out of

$4$ .

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{\frac{2 \cdot 2}{2}} \frac{1}{- 2} d u_{2}$

Step 6.1.4.2

Cancel the common factors.

Tap for more steps...

Step 6.1.4.2.1

Factor

$2$ out of

$2$ .

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{\frac{2 \cdot 2}{2 (1)}} \frac{1}{- 2} d u_{2}$

Step 6.1.4.2.2

Cancel the common factor.

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{\frac{2 \cdot 2}{2 \cdot 1}} \frac{1}{- 2} d u_{2}$

Step 6.1.4.2.3

Rewrite the expression.

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{\frac{2}{1}} \frac{1}{- 2} d u_{2}$

Step 6.1.4.2.4

Divide

$2$ by

$1$ .

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{2} \frac{1}{- 2} d u_{2}$

Step 6.2

Move the negative in front of the fraction.

$\int_{1}^{0} {u_{2}}^{5} {(- u_{2} + 1)}^{2} (- \frac{1}{2}) d u_{2}$

Step 6.3

Combine

${u_{2}}^{5}$ and

$\frac{1}{2}$ .

$\int_{1}^{0} {(- u_{2} + 1)}^{2} (- \frac{{u_{2}}^{5}}{2}) d u_{2}$

Step 6.4

Combine

${(- u_{2} + 1)}^{2}$ and

$\frac{{u_{2}}^{5}}{2}$ .

$\int_{1}^{0} - \frac{{(- u_{2} + 1)}^{2} {u_{2}}^{5}}{2} d u_{2}$

Step 7

Since

$- 1$ is constant with respect to

$u_{2}$ , move

$- 1$ out of the integral.

$- \int_{1}^{0} \frac{{(- u_{2} + 1)}^{2} {u_{2}}^{5}}{2} d u_{2}$

Step 8

Since

$\frac{1}{2}$ is constant with respect to

$u_{2}$ , move

$\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int_{1}^{0} {(- u_{2} + 1)}^{2} {u_{2}}^{5} d u_{2})$

Step 9

Expand

${(- u_{2} + 1)}^{2} {u_{2}}^{5}$ .

Tap for more steps...

Step 9.1

Rewrite

${(- u_{2} + 1)}^{2}$ as

$(- u_{2} + 1) (- u_{2} + 1)$ .

$- \frac{1}{2} \int_{1}^{0} (- u_{2} + 1) (- u_{2} + 1) {u_{2}}^{5} d u_{2}$

Step 9.2