Calculus Examples

$\int_{0}^{\frac{π}{6}} (\sec (θ) \tan (θ)) d θ$

Step 1

Remove parentheses.

$\int_{0}^{\frac{π}{6}} \sec (θ) \tan (θ) d θ$

Step 2

Since the derivative of $\sec (θ)$ is $\sec (θ) \tan (θ)$ , the integral of $\sec (θ) \tan (θ)$ is $\sec (θ)$ .

$\sec (θ)]_{0}^{\frac{π}{6}}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\sec (θ)$ at $\frac{π}{6}$ and at $0$ .

$\sec (\frac{π}{6}) - \sec (0)$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{2}{\sqrt{3}} - \sec (0)$

Step 3.2.2

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

$\frac{2}{\sqrt{3}} - 1 \cdot 1$

Step 3.2.3

Multiply $- 1$ by $1$ .

$\frac{2}{\sqrt{3}} - 1$

Step 3.3

Simplify each term.

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Step 3.3.1

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

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - 1$

Step 3.3.2

Combine and simplify the denominator.

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Step 3.3.2.1

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

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} - 1$

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.2.4

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - 1$

Step 3.3.2.5

Add $1$ and $1$ .

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

Step 3.3.2.6

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

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Step 3.3.2.6.1

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

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

Step 3.3.2.6.2

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

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

Step 3.3.2.6.3

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

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

Step 3.3.2.6.4

Cancel the common factor of $2$ .

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Step 3.3.2.6.4.1

Cancel the common factor.

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

Step 3.3.2.6.4.2

Rewrite the expression.

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

Step 3.3.2.6.5

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3} - 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{3} - 1$

Decimal Form:

$0.15470053 \dots$