Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\cos (x)$ .

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

$u_{lower} = \cos (\frac{π}{4})$

Step 1.3

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

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

Step 1.4

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

$u_{upper} = \cos (\frac{π}{3})$

Step 1.5

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

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

Step 1.6

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

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

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

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $\frac{1}{2}$ by ${(\frac{1}{2})}^{2}$ by adding the exponents.

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Step 5.1

Multiply $\frac{1}{2}$ by ${(\frac{1}{2})}^{2}$ .

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Step 5.1.1

Raise $\frac{1}{2}$ to the power of $1$ .

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

Step 5.1.2

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

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

Step 5.2

Add $1$ and $2$ .

$- ({(\frac{1}{2})}^{3} - \frac{1}{2} {(\frac{\sqrt{2}}{2})}^{2})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- ({(\frac{1}{2})}^{3} - \frac{1}{2} \cdot {(\frac{\sqrt{2}}{2})}^{2})$

Decimal Form:

$0.125$