Calculus Examples

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

Step 1

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

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $4$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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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 $u = \sin (x)$ .

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

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 $u = \sin (x)$ .

$u_{upper} = \sin (\frac{π}{2})$

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 $u$ , $d u$ , and the new limits of integration.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Move $u^{2}$ .

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

Step 5.6

Move $u^{2}$ .

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

Step 5.7

Multiply $1$ by $1$ .

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

Step 5.8

Multiply $- 1$ by $1$ .

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

Step 5.9

Multiply $- 1$ by $1$ .

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

Step 5.10

Multiply $- 1$ by $- 1$ .

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

Step 5.11

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

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

Step 5.12

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

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

Step 5.13

Add $2$ and $2$ .

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

Step 5.14

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

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

Step 5.15

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

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

Step 5.16

Move $1$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Apply the constant rule.

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

Step 12

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

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

Step 13

Substitute and simplify.

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

Simplify.

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Step 13.3.1

One to any power is one.

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

Step 13.3.2

Multiply $\frac{1}{5}$ by $1$ .

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

Step 13.3.3

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

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

Step 13.3.4

Combine the numerators over the common denominator.

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

Step 13.3.5

Add $1$ and $5$ .

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

Step 13.3.6

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

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

Step 13.3.7

Multiply $\frac{1}{5}$ by $0$ .

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

Step 13.3.8

Add $0$ and $0$ .

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

Step 13.3.9

Multiply $- 1$ by $0$ .

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

Step 13.3.10

Add $\frac{6}{5}$ and $0$ .

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

Step 13.3.11

One to any power is one.

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

Step 13.3.12

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

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

Step 13.3.13

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

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Step 13.3.13.1

Factor $3$ out of $0$ .

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

Step 13.3.13.2

Cancel the common factors.

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Step 13.3.13.2.1

Factor $3$ out of $3$ .

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

Step 13.3.13.2.2

Cancel the common factor.

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

Step 13.3.13.2.3

Rewrite the expression.

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

Step 13.3.13.2.4

Divide $0$ by $1$ .

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

Step 13.3.14

Multiply $- 1$ by $0$ .

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

Step 13.3.15

Add $\frac{1}{3}$ and $0$ .

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

Step 13.3.16

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

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

Step 13.3.17

Move the negative in front of the fraction.

$\frac{6}{5} - \frac{2}{3}$

Step 13.3.18

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 13.3.19

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 13.3.20

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

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Step 13.3.20.1

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

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

Step 13.3.20.2

Multiply $5$ by $3$ .

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

Step 13.3.20.3

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

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

Step 13.3.20.4

Multiply $3$ by $5$ .

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

Step 13.3.21

Combine the numerators over the common denominator.

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

Step 13.3.22

Simplify the numerator.

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Step 13.3.22.1

Multiply $6$ by $3$ .

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

Step 13.3.22.2

Multiply $- 2$ by $5$ .

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

Step 13.3.22.3

Subtract $10$ from $18$ .

$\frac{8}{15}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{15}$

Decimal Form:

$0.5\overline{3}$