Calculus Examples

$\sqrt{2} \int_{- \frac{π}{6}}^{\frac{π}{6}} \sec^{2} (y) d y$

Step 1

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

$\sqrt{2} \tan (y)]_{- \frac{π}{6}}^{\frac{π}{6}}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $\tan (y)$ at $\frac{π}{6}$ and at $- \frac{π}{6}$ .

$\sqrt{2} (\tan (\frac{π}{6}) - \tan (- \frac{π}{6}))$

Step 2.2

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

$\sqrt{2} (\frac{\sqrt{3}}{3} - \tan (- \frac{π}{6}))$

Step 2.3

Simplify.

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Step 2.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sqrt{2} (\frac{\sqrt{3}}{3} - \tan (\frac{11 π}{6}))$

Step 2.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$\sqrt{2} (\frac{\sqrt{3}}{3} - - \tan (\frac{π}{6}))$

Step 2.3.3

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

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

Step 2.3.4

Multiply $- - \frac{\sqrt{3}}{3}$ .

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4.2

Multiply $\frac{\sqrt{3}}{3}$ by $1$ .

$\sqrt{2} (\frac{\sqrt{3}}{3} + \frac{\sqrt{3}}{3})$

Step 2.3.5

Combine the numerators over the common denominator.

$\sqrt{2} \frac{\sqrt{3} + \sqrt{3}}{3}$

Step 2.3.6

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

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

Step 2.3.7

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

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Step 2.3.7.1

Combine $\sqrt{2}$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{\sqrt{2} (2 \sqrt{3})}{3}$

Step 2.3.7.2

Combine using the product rule for radicals.

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

Step 2.3.7.3

Multiply $2$ by $3$ .

$\frac{2 \sqrt{6}}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{6}}{3}$

Decimal Form:

$1.63299316 \dots$