Calculus Examples

$\int_{0}^{\frac{π}{6}} 9 \sec^{2} (x) d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int_{0}^{\frac{π}{6}} \sec^{2} (x) d x$

Step 2

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

$9 (\tan (x)]_{0}^{\frac{π}{6}})$

Step 3

Simplify the answer.

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Step 3.1

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

$9 (\tan (\frac{π}{6}) - \tan (0))$

Step 3.2

Simplify.

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Step 3.2.1

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

$9 (\frac{\sqrt{3}}{3} - \tan (0))$

Step 3.2.2

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

$9 (\frac{\sqrt{3}}{3} - 0)$

Step 3.2.3

Multiply $- 1$ by $0$ .

$9 (\frac{\sqrt{3}}{3} + 0)$

Step 3.2.4

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

$9 \frac{\sqrt{3}}{3}$

Step 3.2.5

Combine $9$ and $\frac{\sqrt{3}}{3}$ .

$\frac{9 \sqrt{3}}{3}$

Step 3.2.6

Cancel the common factor of $9$ and $3$ .

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Step 3.2.6.1

Factor $3$ out of $9 \sqrt{3}$ .

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

Step 3.2.6.2

Cancel the common factors.

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Step 3.2.6.2.1

Factor $3$ out of $3$ .

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

Step 3.2.6.2.2

Cancel the common factor.

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

Step 3.2.6.2.3

Rewrite the expression.

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

Step 3.2.6.2.4

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

$3 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{3}$

Decimal Form:

$5.19615242 \dots$