Calculus Examples

$\int_{\frac{π}{6}}^{\frac{π}{2}} \frac{\cos (x)}{3 + \sin (x)} d x$

Step 1

Let $u = 3 + \sin (x)$ . 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 $u = 3 + \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 + \sin (x)$ .

$\frac{d}{d x} [3 + \sin (x)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [\sin (x)]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [\sin (x)]$

Step 1.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\sin (x)]$

Step 1.1.3

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

$0 + \cos (x)$

Step 1.1.4

Add $0$ and $\cos (x)$ .

$\cos (x)$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 + \sin (x)$ .

$u_{lower} = 3 + \sin (\frac{π}{6})$

Step 1.3

Simplify.

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Step 1.3.1

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$u_{lower} = 3 + \frac{1}{2}$

Step 1.3.2

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

$u_{lower} = 3 \cdot \frac{2}{2} + \frac{1}{2}$

Step 1.3.3

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

$u_{lower} = \frac{3 \cdot 2}{2} + \frac{1}{2}$

Step 1.3.4

Combine the numerators over the common denominator.

$u_{lower} = \frac{3 \cdot 2 + 1}{2}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Multiply $3$ by $2$ .

$u_{lower} = \frac{6 + 1}{2}$

Step 1.3.5.2

Add $6$ and $1$ .

$u_{lower} = \frac{7}{2}$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 + \sin (x)$ .

$u_{upper} = 3 + \sin (\frac{π}{2})$

Step 1.5

Simplify.

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Step 1.5.1

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

$u_{upper} = 3 + 1$

Step 1.5.2

Add $3$ and $1$ .

$u_{upper} = 4$

Step 1.6

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

$u_{lower} = \frac{7}{2}$

$u_{upper} = 4$

Step 1.7

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

$\int_{\frac{7}{2}}^{4} \frac{1}{u} d u$

Step 2

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\ln (| u |)]_{\frac{7}{2}}^{4}$

Step 3

Evaluate $\ln (| u |)$ at $4$ and at $\frac{7}{2}$ .

$\ln (| 4 |) - \ln (| \frac{7}{2} |)$

Step 4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| 4 |}{| \frac{7}{2} |})$

Step 5

Simplify.

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Step 5.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\ln (\frac{4}{| \frac{7}{2} |})$

Step 5.2

$\frac{7}{2}$ is approximately $3.5$ which is positive so remove the absolute value

$\ln (\frac{4}{\frac{7}{2}})$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

$\ln (4 (\frac{2}{7}))$

Step 5.4

Multiply $4 (\frac{2}{7})$ .

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Step 5.4.1

Combine $4$ and $\frac{2}{7}$ .

$\ln (\frac{4 \cdot 2}{7})$

Step 5.4.2

Multiply $4$ by $2$ .

$\ln (\frac{8}{7})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{8}{7})$

Decimal Form:

$0.13353139 \dots$