Calculus Examples

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

Step 1

Factor out $\sin^{4} (x)$ .

$\int_{0}^{\frac{π}{2}} \sin^{4} (x) \sin (x) d x$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $4$ .

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

Step 2.2

Rewrite ${\sin (x)}^{2 (2)}$ as exponentiation.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $\cos (x)$ .

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

Step 4.1.2

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

$- \sin (x)$

Step 4.2

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

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

Step 4.3

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

$u_{lower} = 1$

Step 4.4

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

$u_{upper} = \cos (\frac{π}{2})$

Step 4.5

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

$u_{upper} = 0$

Step 4.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 4.7

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

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

Step 5

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

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

Step 6

Expand ${(1 - u^{2})}^{2}$ .

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Step 6.1

Rewrite ${(1 - u^{2})}^{2}$ as $(1 - u^{2}) (1 - u^{2})$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Move $u^{2}$ .

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

Step 6.6

Move $u^{2}$ .

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

Step 6.7

Multiply $1$ by $1$ .

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

Step 6.8

Multiply $- 1$ by $1$ .

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

Step 6.9

Multiply $- 1$ by $1$ .

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

Step 6.10

Multiply $- 1$ by $- 1$ .

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

Step 6.11

Multiply $u^{2}$ by $1$ .

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

Step 6.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.13

Add $2$ and $2$ .

$- \int_{1}^{0} 1 - u^{2} - u^{2} + u^{4} d u$

Step 6.14

Subtract $u^{2}$ from $- u^{2}$ .

$- \int_{1}^{0} 1 - 2 u^{2} + u^{4} d u$

Step 6.15

Reorder $- 2 u^{2}$ and $u^{4}$ .

$- \int_{1}^{0} 1 + u^{4} - 2 u^{2} d u$

Step 6.16

Move $1$ .

$- \int_{1}^{0} u^{4} - 2 u^{2} + 1 d u$

Step 7

Split the single integral into multiple integrals.

$- (\int_{1}^{0} u^{4} d u + \int_{1}^{0} - 2 u^{2} d u + \int_{1}^{0} d u)$

Step 8

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

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

Step 9

Combine $\frac{1}{5}$ and $u^{5}$ .

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

Step 10

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

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

Step 11

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

$- (\frac{u^{5}}{5}]_{1}^{0} - 2 (\frac{1}{3} u^{3}]_{1}^{0}) + \int_{1}^{0} d u)$

Step 12

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

$- (\frac{u^{5}}{5}]_{1}^{0} - 2 (\frac{u^{3}}{3}]_{1}^{0}) + \int_{1}^{0} d u)$

Step 13

Apply the constant rule.

$- (\frac{u^{5}}{5}]_{1}^{0} - 2 (\frac{u^{3}}{3}]_{1}^{0}) + u]_{1}^{0})$

Step 14

Combine $\frac{u^{5}}{5}]_{1}^{0}$ and $u]_{1}^{0}$ .

$- (\frac{u^{5}}{5} + u]_{1}^{0} - 2 (\frac{u^{3}}{3}]_{1}^{0}))$

Step 15

Substitute and simplify.

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Step 15.1

Evaluate $\frac{u^{5}}{5} + u$ at $0$ and at $1$ .

$- ((\frac{0^{5}}{5} + 0) - (\frac{1^{5}}{5} + 1) - 2 (\frac{u^{3}}{3}]_{1}^{0}))$

Step 15.2

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

$- ((\frac{0^{5}}{5} + 0) - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3

Simplify.

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Step 15.3.1

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

$- (\frac{0}{5} + 0 - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.2

Cancel the common factor of $0$ and $5$ .

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Step 15.3.2.1

Factor $5$ out of $0$ .

$- (\frac{5 (0)}{5} + 0 - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.2.2

Cancel the common factors.

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Step 15.3.2.2.1

Factor $5$ out of $5$ .

$- (\frac{5 \cdot 0}{5 \cdot 1} + 0 - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.2.2.2

Cancel the common factor.

$- (\frac{5 \cdot 0}{5 \cdot 1} + 0 - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.2.2.3

Rewrite the expression.

$- (\frac{0}{1} + 0 - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.2.2.4

Divide $0$ by $1$ .

$- (0 + 0 - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.3

Add $0$ and $0$ .

$- (0 - (\frac{1^{5}}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.4

One to any power is one.

$- (0 - (\frac{1}{5} + 1) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.5

Write $1$ as a fraction with a common denominator.

$- (0 - (\frac{1}{5} + \frac{5}{5}) - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.6

Combine the numerators over the common denominator.

$- (0 - \frac{1 + 5}{5} - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.7

Add $1$ and $5$ .

$- (0 - \frac{6}{5} - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.8

Subtract $\frac{6}{5}$ from $0$ .

$- (- \frac{6}{5} - 2 ((\frac{0^{3}}{3}) - \frac{1^{3}}{3}))$

Step 15.3.9

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

$- (- \frac{6}{5} - 2 (\frac{0}{3} - \frac{1^{3}}{3}))$

Step 15.3.10

Cancel the common factor of $0$ and $3$ .

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Step 15.3.10.1

Factor $3$ out of $0$ .

$- (- \frac{6}{5} - 2 (\frac{3 (0)}{3} - \frac{1^{3}}{3}))$

Step 15.3.10.2

Cancel the common factors.

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Step 15.3.10.2.1

Factor $3$ out of $3$ .

$- (- \frac{6}{5} - 2 (\frac{3 \cdot 0}{3 \cdot 1} - \frac{1^{3}}{3}))$

Step 15.3.10.2.2

Cancel the common factor.

$- (- \frac{6}{5} - 2 (\frac{3 \cdot 0}{3 \cdot 1} - \frac{1^{3}}{3}))$

Step 15.3.10.2.3

Rewrite the expression.

$- (- \frac{6}{5} - 2 (\frac{0}{1} - \frac{1^{3}}{3}))$

Step 15.3.10.2.4

Divide $0$ by $1$ .

$- (- \frac{6}{5} - 2 (0 - \frac{1^{3}}{3}))$

Step 15.3.11

One to any power is one.

$- (- \frac{6}{5} - 2 (0 - \frac{1}{3}))$

Step 15.3.12

Subtract $\frac{1}{3}$ from $0$ .

$- (- \frac{6}{5} - 2 (- \frac{1}{3}))$

Step 15.3.13

Multiply $- 1$ by $- 2$ .

$- (- \frac{6}{5} + 2 (\frac{1}{3}))$

Step 15.3.14

Combine $2$ and $\frac{1}{3}$ .

$- (- \frac{6}{5} + \frac{2}{3})$

Step 15.3.15

To write $- \frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- (- \frac{6}{5} \cdot \frac{3}{3} + \frac{2}{3})$

Step 15.3.16

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- (- \frac{6}{5} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{5}{5})$

Step 15.3.17

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.3.17.1

Multiply $\frac{6}{5}$ by $\frac{3}{3}$ .

$- (- \frac{6 \cdot 3}{5 \cdot 3} + \frac{2}{3} \cdot \frac{5}{5})$

Step 15.3.17.2

Multiply $5$ by $3$ .

$- (- \frac{6 \cdot 3}{15} + \frac{2}{3} \cdot \frac{5}{5})$

Step 15.3.17.3

Multiply $\frac{2}{3}$ by $\frac{5}{5}$ .

$- (- \frac{6 \cdot 3}{15} + \frac{2 \cdot 5}{3 \cdot 5})$

Step 15.3.17.4

Multiply $3$ by $5$ .

$- (- \frac{6 \cdot 3}{15} + \frac{2 \cdot 5}{15})$

Step 15.3.18

Combine the numerators over the common denominator.

$- \frac{- 6 \cdot 3 + 2 \cdot 5}{15}$

Step 15.3.19

Simplify the numerator.

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Step 15.3.19.1

Multiply $- 6$ by $3$ .

$- \frac{- 18 + 2 \cdot 5}{15}$

Step 15.3.19.2

Multiply $2$ by $5$ .

$- \frac{- 18 + 10}{15}$

Step 15.3.19.3

Add $- 18$ and $10$ .

$- \frac{- 8}{15}$

Step 15.3.20

Move the negative in front of the fraction.

$- - \frac{8}{15}$

Step 15.3.21

Multiply $- 1$ by $- 1$ .

$1 (\frac{8}{15})$

Step 15.3.22

Multiply $\frac{8}{15}$ by $1$ .

$\frac{8}{15}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{15}$

Decimal Form:

$0.5\overline{3}$