Calculus Examples

$\int_{0}^{\frac{π}{12}} 18 \tan (3 x) d x$

Step 1

Since $18$ is constant with respect to $x$ , move $18$ out of the integral.

$18 \int_{0}^{\frac{π}{12}} \tan (3 x) d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $3 x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

$3 \cdot 1$

Step 2.1.4

Multiply $3$ by $1$ .

$3$

Step 2.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot 0$

Step 2.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 \frac{π}{12}$

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Factor $3$ out of $12$ .

$u_{upper} = 3 \frac{π}{3 (4)}$

Step 2.5.2

Cancel the common factor.

$u_{upper} = 3 \frac{π}{3 \cdot 4}$

Step 2.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{4}$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{4}$

Step 2.7

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

$18 \int_{0}^{\frac{π}{4}} \tan (u) \frac{1}{3} d u$

Step 3

Combine $\tan (u)$ and $\frac{1}{3}$ .

$18 \int_{0}^{\frac{π}{4}} \frac{\tan (u)}{3} d u$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$18 (\frac{1}{3} \int_{0}^{\frac{π}{4}} \tan (u) d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{3}$ and $18$ .

$\frac{18}{3} \int_{0}^{\frac{π}{4}} \tan (u) d u$

Step 5.2

Cancel the common factor of $18$ and $3$ .

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Step 5.2.1

Factor $3$ out of $18$ .

$\frac{3 \cdot 6}{3} \int_{0}^{\frac{π}{4}} \tan (u) d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 6}{3 (1)} \int_{0}^{\frac{π}{4}} \tan (u) d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{3 \cdot 6}{3 \cdot 1} \int_{0}^{\frac{π}{4}} \tan (u) d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{6}{1} \int_{0}^{\frac{π}{4}} \tan (u) d u$

Step 5.2.2.4

Divide $6$ by $1$ .

$6 \int_{0}^{\frac{π}{4}} \tan (u) d u$

Step 6

The integral of $\tan (u)$ with respect to $u$ is $\ln (| \sec (u) |)$ .

$6 (\ln (| \sec (u) |)]_{0}^{\frac{π}{4}})$

Step 7

Evaluate $\ln (| \sec (u) |)$ at $\frac{π}{4}$ and at $0$ .

$6 (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |))$

Step 8

Simplify.

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Step 8.1

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

$6 (\ln (| \frac{2}{\sqrt{2}} |) - \ln (| \sec (0) |))$

Step 8.2

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

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

Step 8.3

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

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

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 9.1.2

Combine and simplify the denominator.

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Step 9.1.2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 9.1.2.2

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

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

Step 9.1.2.3

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

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

Step 9.1.2.4

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

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

Step 9.1.2.5

Add $1$ and $1$ .

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

Step 9.1.2.6

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

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Step 9.1.2.6.1

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

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

Step 9.1.2.6.2

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

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

Step 9.1.2.6.3

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

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

Step 9.1.2.6.4

Cancel the common factor of $2$ .

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Step 9.1.2.6.4.1

Cancel the common factor.

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

Step 9.1.2.6.4.2

Rewrite the expression.

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

Step 9.1.2.6.5

Evaluate the exponent.

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

Step 9.1.3

Cancel the common factor of $2$ .

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Step 9.1.3.1

Cancel the common factor.

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

Step 9.1.3.2

Divide $\sqrt{2}$ by $1$ .

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

Step 9.1.4

$\sqrt{2}$ is approximately $1.41421356$ which is positive so remove the absolute value

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

Step 9.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

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

Step 9.3

Divide $\sqrt{2}$ by $1$ .

$6 \ln (\sqrt{2})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$6 \ln (\sqrt{2})$

Decimal Form:

$2.07944154 \dots$