Calculus Examples

$\int_{0}^{\frac{π}{4}} \sin^{5} (2 x) d x$

Step 1

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

Substitute the lower limit in for $x$ in $u_{1} = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 1.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{1} = 2 x$ .

$u_{upper} = 2 \frac{π}{4}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $4$ .

$u_{upper} = 2 \frac{π}{2 (2)}$

Step 1.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 2}$

Step 1.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 1.7

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

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

Step 2

Combine $\sin^{5} (u_{1})$ and $\frac{1}{2}$ .

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

Step 3

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

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

Step 4

Factor out $\sin^{4} (u_{1})$ .

$\frac{1}{2} \int_{0}^{\frac{π}{2}} \sin^{4} (u_{1}) \sin (u_{1}) d u_{1}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $2$ out of $4$ .

$\frac{1}{2} \int_{0}^{\frac{π}{2}} {\sin (u_{1})}^{2 (2)} \sin (u_{1}) d u_{1}$

Step 5.2

Rewrite ${\sin (u_{1})}^{2 (2)}$ as exponentiation.

$\frac{1}{2} \int_{0}^{\frac{π}{2}} {(\sin^{2} (u_{1}))}^{2} \sin (u_{1}) d u_{1}$

Step 6

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

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

Step 7

Let $u_{2} = \cos (u_{1})$ . Then $d u_{2} = - \sin (u_{1}) d u_{1}$ , so $- \frac{1}{\sin (u_{1})} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.1

Let $u_{2} = \cos (u_{1})$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 7.1.1

Differentiate $\cos (u_{1})$ .

$\frac{d}{d u_{1}} [\cos (u_{1})]$

Step 7.1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1})$

Step 7.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = \cos (u_{1})$ .

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

Step 7.3

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

$u_{lower} = 1$

Step 7.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = \cos (u_{1})$ .

$u_{upper} = \cos (\frac{π}{2})$

Step 7.5

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

$u_{upper} = 0$

Step 7.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 7.7

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

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

Step 8

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 9.4

Apply the distributive property.

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

Step 9.5

Move ${u_{2}}^{2}$ .

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

Step 9.6

Move ${u_{2}}^{2}$ .

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

Step 9.7

Multiply $1$ by $1$ .

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

Step 9.8

Multiply $- 1$ by $1$ .

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

Step 9.9

Multiply $- 1$ by $1$ .

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

Step 9.10

Multiply $- 1$ by $- 1$ .

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

Step 9.11

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

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

Step 9.12

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

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

Step 9.13

Add $2$ and $2$ .

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

Step 9.14

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

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

Step 9.15

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

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

Step 9.16

Move $1$ .

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

Step 10

Split the single integral into multiple integrals.

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

Step 11

By the Power Rule, the integral of ${u_{2}}^{4}$ with respect to $u_{2}$ is $\frac{1}{5} {u_{2}}^{5}$ .

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

Step 12

Combine $\frac{1}{5}$ and ${u_{2}}^{5}$ .

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Apply the constant rule.

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

Step 17

Combine $\frac{{u_{2}}^{5}}{5}]_{1}^{0}$ and $u_{2}]_{1}^{0}$ .

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

Step 18

Substitute and simplify.

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Step 18.1

Evaluate $\frac{{u_{2}}^{5}}{5} + u_{2}$ at $0$ and at $1$ .

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

Step 18.2

Evaluate $\frac{{u_{2}}^{3}}{3}$ at $0$ and at $1$ .

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

Step 18.3

Simplify.

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Step 18.3.1

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

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

Step 18.3.2

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

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Step 18.3.2.1

Factor $5$ out of $0$ .

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

Step 18.3.2.2

Cancel the common factors.

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Step 18.3.2.2.1

Factor $5$ out of $5$ .

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

Step 18.3.2.2.2

Cancel the common factor.

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

Step 18.3.2.2.3

Rewrite the expression.

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

Step 18.3.2.2.4

Divide $0$ by $1$ .

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

Step 18.3.3

Add $0$ and $0$ .

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

Step 18.3.4

One to any power is one.

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

Step 18.3.5

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

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

Step 18.3.6

Combine the numerators over the common denominator.

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

Step 18.3.7

Add $1$ and $5$ .

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

Step 18.3.8

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

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

Step 18.3.9

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

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

Step 18.3.10

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

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Step 18.3.10.1

Factor $3$ out of $0$ .

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

Step 18.3.10.2

Cancel the common factors.

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Step 18.3.10.2.1

Factor $3$ out of $3$ .

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

Step 18.3.10.2.2

Cancel the common factor.

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

Step 18.3.10.2.3

Rewrite the expression.

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

Step 18.3.10.2.4

Divide $0$ by $1$ .

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

Step 18.3.11

One to any power is one.

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

Step 18.3.12

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

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

Step 18.3.13

Multiply $- 1$ by $- 2$ .

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

Step 18.3.14

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

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

Step 18.3.15

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

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

Step 18.3.16

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

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

Step 18.3.17

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

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Step 18.3.17.1

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

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

Step 18.3.17.2

Multiply $5$ by $3$ .

$\frac{1}{2} (- (- \frac{6 \cdot 3}{15} + \frac{2}{3} \cdot \frac{5}{5}))$

Step 18.3.17.3

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

$\frac{1}{2} (- (- \frac{6 \cdot 3}{15} + \frac{2 \cdot 5}{3 \cdot 5}))$

Step 18.3.17.4

Multiply $3$ by $5$ .

$\frac{1}{2} (- (- \frac{6 \cdot 3}{15} + \frac{2 \cdot 5}{15}))$

Step 18.3.18

Combine the numerators over the common denominator.

$\frac{1}{2} (- \frac{- 6 \cdot 3 + 2 \cdot 5}{15})$

Step 18.3.19

Simplify the numerator.

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Step 18.3.19.1

Multiply $- 6$ by $3$ .

$\frac{1}{2} (- \frac{- 18 + 2 \cdot 5}{15})$

Step 18.3.19.2

Multiply $2$ by $5$ .

$\frac{1}{2} (- \frac{- 18 + 10}{15})$

Step 18.3.19.3

Add $- 18$ and $10$ .

$\frac{1}{2} (- \frac{- 8}{15})$

Step 18.3.20

Move the negative in front of the fraction.

$\frac{1}{2} (- - \frac{8}{15})$

Step 18.3.21

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} (1 (\frac{8}{15}))$

Step 18.3.22

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

$\frac{1}{2} \cdot \frac{8}{15}$

Step 18.3.23

Multiply $\frac{1}{2}$ by $\frac{8}{15}$ .

$\frac{8}{2 \cdot 15}$

Step 18.3.24

Multiply $2$ by $15$ .

$\frac{8}{30}$

Step 18.3.25

Cancel the common factor of $8$ and $30$ .

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Step 18.3.25.1

Factor $2$ out of $8$ .

$\frac{2 (4)}{30}$

Step 18.3.25.2

Cancel the common factors.

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Step 18.3.25.2.1

Factor $2$ out of $30$ .

$\frac{2 \cdot 4}{2 \cdot 15}$

Step 18.3.25.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 15}$

Step 18.3.25.2.3

Rewrite the expression.

$\frac{4}{15}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{15}$

Decimal Form:

$0.2\overline{6}$