Calculus Examples

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

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $\cos (x)$ .

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

Step 2.1.2

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

$- \sin (x)$

Step 2.2

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

$u_{lower} = \cos (\frac{π}{3})$

Step 2.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$u_{lower} = \frac{1}{2}$

Step 2.4

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

$u_{upper} = \cos (2 π)$

Step 2.5

Simplify.

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Step 2.5.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 2.5.2

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

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = \frac{1}{2}$

$u_{upper} = 1$

Step 2.7

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

$4 \int_{\frac{1}{2}}^{1} - u^{3} d u$

Step 3

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

$4 (- \int_{\frac{1}{2}}^{1} u^{3} d u)$

Step 4

Multiply $- 1$ by $4$ .

$- 4 \int_{\frac{1}{2}}^{1} u^{3} d u$

Step 5

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

$- 4 (\frac{1}{4} u^{4}]_{\frac{1}{2}}^{1})$

Step 6

Combine $\frac{1}{4}$ and $u^{4}$ .

$- 4 (\frac{u^{4}}{4}]_{\frac{1}{2}}^{1})$

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

One to any power is one.

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

Step 8

Simplify.

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Step 8.1

Combine the numerators over the common denominator.

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

Step 8.2

Simplify each term.

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Step 8.2.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 8.2.2

One to any power is one.

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

Step 8.2.3

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

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

Step 8.3

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

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

Step 8.4

Combine the numerators over the common denominator.

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

Step 8.5

Subtract $1$ from $16$ .

$- 4 \frac{\frac{15}{16}}{4}$

Step 8.6

Cancel the common factor of $4$ .

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Step 8.6.1

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{\frac{15}{16}}{4}$

Step 8.6.2

Cancel the common factor.

$4 \cdot - 1 \frac{\frac{15}{16}}{4}$

Step 8.6.3

Rewrite the expression.

$- \frac{15}{16}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{15}{16}$

Decimal Form:

$- 0.9375$