Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{2}} \sin^{3} (x) \cos (x) d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\sin (x)$ .

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

Step 1.1.2

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

$\cos (x)$

Step 1.2

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

$u_{lower} = \sin (\frac{π}{4})$

Step 1.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$u_{lower} = \frac{\sqrt{2}}{2}$

Step 1.4

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

$u_{upper} = \sin (\frac{π}{2})$

Step 1.5

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = \frac{\sqrt{2}}{2}$

$u_{upper} = 1$

Step 1.7

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

$\int_{\frac{\sqrt{2}}{2}}^{1} u^{3} d u$

Step 2

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{4} u^{4}]_{\frac{\sqrt{2}}{2}}^{1}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{4} u^{4}$ at $1$ and at $\frac{\sqrt{2}}{2}$ .

$(\frac{1}{4} \cdot 1^{4}) - \frac{1}{4} {(\frac{\sqrt{2}}{2})}^{4}$

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$\frac{1}{4} \cdot 1 - \frac{1}{4} {(\frac{\sqrt{2}}{2})}^{4}$

Step 3.2.2

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4} - \frac{1}{4} {(\frac{\sqrt{2}}{2})}^{4}$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$\frac{1}{4} - \frac{1}{4} \cdot \frac{{\sqrt{2}}^{4}}{2^{4}}$

Step 4.1.2

Simplify the numerator.

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Step 4.1.2.1

Rewrite ${\sqrt{2}}^{4}$ as $2^{2}$ .

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Step 4.1.2.1.1

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

$\frac{1}{4} - \frac{1}{4} \cdot \frac{{(2^{\frac{1}{2}})}^{4}}{2^{4}}$

Step 4.1.2.1.2

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

$\frac{1}{4} - \frac{1}{4} \cdot \frac{2^{\frac{1}{2} \cdot 4}}{2^{4}}$

Step 4.1.2.1.3

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

$\frac{1}{4} - \frac{1}{4} \cdot \frac{2^{\frac{4}{2}}}{2^{4}}$

Step 4.1.2.1.4

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

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Step 4.1.2.1.4.1

Factor $2$ out of $4$ .

$\frac{1}{4} - \frac{1}{4} \cdot \frac{2^{\frac{2 \cdot 2}{2}}}{2^{4}}$

Step 4.1.2.1.4.2

Cancel the common factors.

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Step 4.1.2.1.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{4} - \frac{1}{4} \cdot \frac{2^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}}$

Step 4.1.2.1.4.2.2

Cancel the common factor.

$\frac{1}{4} - \frac{1}{4} \cdot \frac{2^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}}$

Step 4.1.2.1.4.2.3

Rewrite the expression.

$\frac{1}{4} - \frac{1}{4} \cdot \frac{2^{\frac{2}{1}}}{2^{4}}$

Step 4.1.2.1.4.2.4

Divide $2$ by $1$ .

$\frac{1}{4} - \frac{1}{4} \cdot \frac{2^{2}}{2^{4}}$

Step 4.1.2.2

Raise $2$ to the power of $2$ .

$\frac{1}{4} - \frac{1}{4} \cdot \frac{4}{2^{4}}$

Step 4.1.3

Raise $2$ to the power of $4$ .

$\frac{1}{4} - \frac{1}{4} \cdot \frac{4}{16}$

Step 4.1.4

Cancel the common factor of $4$ .

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Step 4.1.4.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$\frac{1}{4} + \frac{- 1}{4} \cdot \frac{4}{16}$

Step 4.1.4.2

Cancel the common factor.

$\frac{1}{4} + \frac{- 1}{4} \cdot \frac{4}{16}$

Step 4.1.4.3

Rewrite the expression.

$\frac{1}{4} - \frac{1}{16}$

Step 4.2

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} \cdot \frac{4}{4} - \frac{1}{16}$

Step 4.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\frac{4}{4 \cdot 4} - \frac{1}{16}$

Step 4.3.2

Multiply $4$ by $4$ .

$\frac{4}{16} - \frac{1}{16}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{4 - 1}{16}$

Step 4.5

Subtract $1$ from $4$ .

$\frac{3}{16}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{16}$

Decimal Form:

$0.1875$