Calculus Examples

$\int_{0}^{\frac{π}{2}} \cos^{9} (x) \sin^{5} (x) d x$

Step 1

Factor out $\cos^{8} (x)$ .

$\int_{0}^{\frac{π}{2}} \cos^{8} (x) \cos (x) \sin^{5} (x) d x$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $8$ .

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

Step 2.2

Rewrite ${\cos (x)}^{2 (4)}$ as exponentiation.

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

Step 3

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

$\int_{0}^{\frac{π}{2}} {(1 - \sin^{2} (x))}^{4} \cos (x) \sin^{5} (x) d x$

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 $d$ $u_{1}$ .

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Step 4.1

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

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

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

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.

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

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 $d$ $u_{2}$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Differentiate.

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

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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.

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Step 5.3.1

Simplify each term.

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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.

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Step 5.5.1

Simplify each term.

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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.

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

Step 6

Simplify.

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Step 6.1

Rewrite ${\sqrt{- u_{2} + 1}}^{4}$ as ${(- u_{2} + 1)}^{2}$ .

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Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- u_{2} + 1}$ as ${(- u_{2} + 1)}^{\frac{1}{2}}$ .

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

Step 6.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.1.3

Combine $\frac{1}{2}$ and $4$ .

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

Step 6.1.4

Cancel the common factor of $4$ and $2$ .

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Step 6.1.4.1

Factor $2$ out of $4$ .

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

Step 6.1.4.2

Cancel the common factors.

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Step 6.1.4.2.1

Factor $2$ out of $2$ .

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

Step 6.1.4.2.2

Cancel the common factor.

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

Step 6.1.4.2.3

Rewrite the expression.

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

Step 6.1.4.2.4

Divide $2$ by $1$ .

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

Step 6.2

Move the negative in front of the fraction.

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

Step 6.3

Combine ${u_{2}}^{4}$ and $\frac{1}{2}$ .

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

Step 6.4

Combine ${(- u_{2} + 1)}^{2}$ and $\frac{{u_{2}}^{4}}{2}$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Expand ${(- u_{2} + 1)}^{2} {u_{2}}^{4}$ .

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Step 9.1

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

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

Step 9.2