Calculus Examples

$\int_{0}^{\frac{π}{6}} 3 \sec^{2} (x) d x$

Step 1

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

$3 \int_{0}^{\frac{π}{6}} \sec^{2} (x) d x$

Step 2

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$3 (\tan (x)]_{0}^{\frac{π}{6}})$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\tan (x)$ at $\frac{π}{6}$ and at $0$ .

$3 (\tan (\frac{π}{6}) - \tan (0))$

Step 3.2

Simplify.

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Step 3.2.1

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

$3 (\frac{\sqrt{3}}{3} - \tan (0))$

Step 3.2.2

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

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

Step 3.2.3

Multiply $- 1$ by $0$ .

$3 (\frac{\sqrt{3}}{3} + 0)$

Step 3.2.4

Add $\frac{\sqrt{3}}{3}$ and $0$ .

$3 \frac{\sqrt{3}}{3}$

Step 3.2.5

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

$\frac{3 \sqrt{3}}{3}$

Step 3.2.6

Cancel the common factor of $3$ .

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Step 3.2.6.1

Cancel the common factor.

$\frac{3 \sqrt{3}}{3}$

Step 3.2.6.2

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

$\sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3}$

Decimal Form:

$1.73205080 \dots$