Calculus Examples

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

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

Step 1

Since the derivative of

tan (y)

$\tan (y)$ is

{sec}^{2} (y)

$\sec^{2} (y)$ , the integral of

{sec}^{2} (y)

$\sec^{2} (y)$ is

tan (y)

$\tan (y)$ .

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

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

Step 2

Simplify the answer.

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Step 2.1

Evaluate

tan (y)

$\tan (y)$ at

\frac{π}{6}

$\frac{π}{6}$ and at

- \frac{π}{6}

$- \frac{π}{6}$ .

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

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

Step 2.2

The exact value of

tan (\frac{π}{6})

$\tan (\frac{π}{6})$ is

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

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

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

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

Step 2.3

Simplify.

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Step 2.3.1

Add full rotations of

2 π

$2 π$ until the angle is greater than or equal to

0

$0$ and less than

2 π

$2 π$ .

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

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

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}))

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

Step 2.3.3

The exact value of

tan (\frac{π}{6})

$\tan (\frac{π}{6})$ is

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

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

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

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

Step 2.3.4

Multiply

- - \frac{\sqrt{3}}{3}

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

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Step 2.3.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

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:

$1.63299316 \dots$