Calculus Examples

$\int_{0}^{\frac{π}{6}} \sec (y) d y$

Step 1

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

$\ln (| \sec (y) + \tan (y) |)]_{0}^{\frac{π}{6}}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $\ln (| \sec (y) + \tan (y) |)$ at $\frac{π}{6}$ and at $0$ .

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

Step 2.2

Simplify.

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Step 2.2.1

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\ln (| \frac{2}{\sqrt{3}} + \tan (\frac{π}{6}) |) - \ln (| \sec (0) + \tan (0) |)$

Step 2.2.2

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\ln (| \frac{2}{\sqrt{3}} + \frac{\sqrt{3}}{3} |) - \ln (| \sec (0) + \tan (0) |)$

Step 2.2.3

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

$\ln (| \frac{2}{\sqrt{3}} + \frac{\sqrt{3}}{3} |) - \ln (| 1 + \tan (0) |)$

Step 2.2.4

The exact value of $\tan (0)$ is $0$ .

$\ln (| \frac{2}{\sqrt{3}} + \frac{\sqrt{3}}{3} |) - \ln (| 1 + 0 |)$

Step 2.2.5

Add $1$ and $0$ .

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

Step 2.2.6

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

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

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Simplify each term.

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Step 2.3.1.1.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.3.1.1.2

Combine and simplify the denominator.

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Step 2.3.1.1.2.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.3.1.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.3.1.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.3.1.1.2.4

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

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

Step 2.3.1.1.2.5

Add $1$ and $1$ .

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

Step 2.3.1.1.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.3.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.3.1.1.2.6.2

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

$\ln (\frac{| \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + \frac{\sqrt{3}}{3} |}{| 1 |})$

Step 2.3.1.1.2.6.3

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

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

Step 2.3.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.1.2.6.4.1

Cancel the common factor.

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

Step 2.3.1.1.2.6.4.2

Rewrite the expression.

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

Step 2.3.1.1.2.6.5

Evaluate the exponent.

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

Step 2.3.1.2

Combine the numerators over the common denominator.

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

Step 2.3.1.3

Add $2 \sqrt{3}$ and $\sqrt{3}$ .

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

Step 2.3.1.4

Cancel the common factor of $3$ .

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Step 2.3.1.4.1

Cancel the common factor.

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

Step 2.3.1.4.2

Divide $\sqrt{3}$ by $1$ .

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

Step 2.3.1.5

$\sqrt{3}$ is approximately $1.7320508$ which is positive so remove the absolute value

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

Step 2.3.2

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

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

Step 2.3.3

Divide $\sqrt{3}$ by $1$ .

$\ln (\sqrt{3})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\ln (\sqrt{3})$

Decimal Form:

$0.54930614 \dots$