Calculus Examples

$\int_{\frac{π}{2}}^{π} 2 \cot (\frac{x}{3}) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{\frac{π}{2}}^{π} \cot (\frac{x}{3}) d x$

Step 2

Let $u = \frac{x}{3}$ . Then $d u = \frac{1}{3} d x$ , so $3 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \frac{x}{3}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\frac{x}{3}$ .

$\frac{d}{d x} [\frac{x}{3}]$

Step 2.1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{3} \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{3} \cdot 1$

Step 2.1.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 2.2

Substitute the lower limit in for $x$ in $u = \frac{x}{3}$ .

$u_{lower} = \frac{\frac{π}{2}}{3}$

Step 2.3

Simplify.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

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Step 2.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

$u_{lower} = \frac{π}{2 \cdot 3}$

Step 2.3.2.2

Multiply $2$ by $3$ .

$u_{lower} = \frac{π}{6}$

Step 2.4

Substitute the upper limit in for $x$ in $u = \frac{x}{3}$ .

$u_{upper} = \frac{π}{3}$

Step 2.5

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

$u_{lower} = \frac{π}{6}$

$u_{upper} = \frac{π}{3}$

Step 2.6

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

$2 \int_{\frac{π}{6}}^{\frac{π}{3}} \cot (u) \frac{1}{\frac{1}{3}} d u$

Step 3

Simplify.

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Step 3.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$2 \int_{\frac{π}{6}}^{\frac{π}{3}} \cot (u) (1 \cdot 3) d u$

Step 3.2

Multiply $3$ by $1$ .

$2 \int_{\frac{π}{6}}^{\frac{π}{3}} \cot (u) \cdot 3 d u$

Step 3.3

Move $3$ to the left of $\cot (u)$ .

$2 \int_{\frac{π}{6}}^{\frac{π}{3}} 3 \cot (u) d u$

Step 4

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$2 (3 \int_{\frac{π}{6}}^{\frac{π}{3}} \cot (u) d u)$

Step 5

Multiply $3$ by $2$ .

$6 \int_{\frac{π}{6}}^{\frac{π}{3}} \cot (u) d u$

Step 6

The integral of $\cot (u)$ with respect to $u$ is $\ln (| \sin (u) |)$ .

$6 (\ln (| \sin (u) |)]_{\frac{π}{6}}^{\frac{π}{3}})$

Step 7

Evaluate $\ln (| \sin (u) |)$ at $\frac{π}{3}$ and at $\frac{π}{6}$ .

$6 (\ln (| \sin (\frac{π}{3}) |) - \ln (| \sin (\frac{π}{6}) |))$

Step 8

Simplify.

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Step 8.1

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

$6 (\ln (| \frac{\sqrt{3}}{2} |) - \ln (| \sin (\frac{π}{6}) |))$

Step 8.2

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

$6 (\ln (| \frac{\sqrt{3}}{2} |) - \ln (| \frac{1}{2} |))$

Step 8.3

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

$6 \ln (\frac{| \frac{\sqrt{3}}{2} |}{| \frac{1}{2} |})$

Step 9

Simplify.

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Step 9.1

$\frac{\sqrt{3}}{2}$ is approximately $0.8660254$ which is positive so remove the absolute value

$6 \ln (\frac{\frac{\sqrt{3}}{2}}{| \frac{1}{2} |})$

Step 9.2

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$6 \ln (\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}})$

Step 9.3

Multiply the numerator by the reciprocal of the denominator.

$6 \ln (\frac{\sqrt{3}}{2} \cdot 2)$

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

$6 \ln (\frac{\sqrt{3}}{2} \cdot 2)$

Step 9.4.2

Rewrite the expression.

$6 \ln (\sqrt{3})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$6 \ln (\sqrt{3})$

Decimal Form:

$3.29583686 \dots$