Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Rewrite ${(\sec (x) + \tan (x))}^{2}$ as $(\sec (x) + \tan (x)) (\sec (x) + \tan (x))$ .

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

Step 1.2

Expand $(\sec (x) + \tan (x)) (\sec (x) + \tan (x))$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $\sec (x) \sec (x)$ .

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Step 1.3.1.1.1

Raise $\sec (x)$ to the power of $1$ .

$\int_{0}^{\frac{π}{6}} \sec^{1} (x) \sec (x) + \sec (x) \tan (x) + \tan (x) \sec (x) + \tan (x) \tan (x) d x$

Step 1.3.1.1.2

Raise $\sec (x)$ to the power of $1$ .

$\int_{0}^{\frac{π}{6}} \sec^{1} (x) \sec^{1} (x) + \sec (x) \tan (x) + \tan (x) \sec (x) + \tan (x) \tan (x) d x$

Step 1.3.1.1.3

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

$\int_{0}^{\frac{π}{6}} {\sec (x)}^{1 + 1} + \sec (x) \tan (x) + \tan (x) \sec (x) + \tan (x) \tan (x) d x$

Step 1.3.1.1.4

Add $1$ and $1$ .

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

Step 1.3.1.2

Multiply $\tan (x) \tan (x)$ .

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Step 1.3.1.2.1

Raise $\tan (x)$ to the power of $1$ .

$\int_{0}^{\frac{π}{6}} \sec^{2} (x) + \sec (x) \tan (x) + \tan (x) \sec (x) + \tan^{1} (x) \tan (x) d x$

Step 1.3.1.2.2

Raise $\tan (x)$ to the power of $1$ .

$\int_{0}^{\frac{π}{6}} \sec^{2} (x) + \sec (x) \tan (x) + \tan (x) \sec (x) + \tan^{1} (x) \tan^{1} (x) d x$

Step 1.3.1.2.3

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

$\int_{0}^{\frac{π}{6}} \sec^{2} (x) + \sec (x) \tan (x) + \tan (x) \sec (x) + {\tan (x)}^{1 + 1} d x$

Step 1.3.1.2.4

Add $1$ and $1$ .

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

Step 1.3.2

Reorder the factors of $\tan (x) \sec (x)$ .

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

Step 1.3.3

Add $\sec (x) \tan (x)$ and $\sec (x) \tan (x)$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\tan (x)]_{0}^{\frac{π}{6}} + \int_{0}^{\frac{π}{6}} 2 \sec (x) \tan (x) d x + \int_{0}^{\frac{π}{6}} \tan^{2} (x) d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\tan (x)]_{0}^{\frac{π}{6}} + 2 \int_{0}^{\frac{π}{6}} \sec (x) \tan (x) d x + \int_{0}^{\frac{π}{6}} \tan^{2} (x) d x$

Step 5

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

$\tan (x)]_{0}^{\frac{π}{6}} + 2 (\sec (x)]_{0}^{\frac{π}{6}}) + \int_{0}^{\frac{π}{6}} \tan^{2} (x) d x$

Step 6

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

$\tan (x)]_{0}^{\frac{π}{6}} + 2 (\sec (x)]_{0}^{\frac{π}{6}}) + \int_{0}^{\frac{π}{6}} - 1 + \sec^{2} (x) d x$

Step 7

Split the single integral into multiple integrals.

$\tan (x)]_{0}^{\frac{π}{6}} + 2 (\sec (x)]_{0}^{\frac{π}{6}}) + \int_{0}^{\frac{π}{6}} - 1 d x + \int_{0}^{\frac{π}{6}} \sec^{2} (x) d x$

Step 8

Apply the constant rule.

$\tan (x)]_{0}^{\frac{π}{6}} + 2 (\sec (x)]_{0}^{\frac{π}{6}}) + - x]_{0}^{\frac{π}{6}} + \int_{0}^{\frac{π}{6}} \sec^{2} (x) d x$

Step 9

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

$\tan (x)]_{0}^{\frac{π}{6}} + 2 (\sec (x)]_{0}^{\frac{π}{6}}) + - x]_{0}^{\frac{π}{6}} + \tan (x)]_{0}^{\frac{π}{6}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $- x]_{0}^{\frac{π}{6}}$ and $\tan (x)]_{0}^{\frac{π}{6}}$ .

$\tan (x)]_{0}^{\frac{π}{6}} + 2 (\sec (x)]_{0}^{\frac{π}{6}}) + - x + \tan (x)]_{0}^{\frac{π}{6}}$

Step 10.1.2

Combine $\tan (x)]_{0}^{\frac{π}{6}}$ and $- x + \tan (x)]_{0}^{\frac{π}{6}}$ .

$2 (\sec (x)]_{0}^{\frac{π}{6}}) + \tan (x) - x + \tan (x)]_{0}^{\frac{π}{6}}$

Step 10.1.3

Add $\tan (x)$ and $\tan (x)$ .

$2 (\sec (x)]_{0}^{\frac{π}{6}}) + - x + 2 \tan (x)]_{0}^{\frac{π}{6}}$

Step 10.2

Substitute and simplify.

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Step 10.2.1

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

$2 ((\sec (\frac{π}{6})) - \sec (0)) + - x + 2 \tan (x)]_{0}^{\frac{π}{6}}$

Step 10.2.2

Evaluate $- x + 2 \tan (x)$ at $\frac{π}{6}$ and at $0$ .

$2 ((\sec (\frac{π}{6})) - \sec (0)) + (- \frac{π}{6} + 2 \tan (\frac{π}{6})) - (- 0 + 2 \tan (0))$

Step 10.2.3

Simplify.

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Step 10.2.3.1

Multiply $- 1$ by $0$ .

$2 (\sec (\frac{π}{6}) - \sec (0)) - \frac{π}{6} + 2 \tan (\frac{π}{6}) - (0 + 2 \tan (0))$

Step 10.2.3.2

Add $0$ and $2 \tan (0)$ .

$2 (\sec (\frac{π}{6}) - \sec (0)) - \frac{π}{6} + 2 \tan (\frac{π}{6}) - (2 \tan (0))$

Step 10.2.3.3

Multiply $2$ by $- 1$ .

$2 (\sec (\frac{π}{6}) - \sec (0)) - \frac{π}{6} + 2 \tan (\frac{π}{6}) - 2 \tan (0)$

Step 10.3

Simplify.

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Step 10.3.1

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

$2 (\frac{2}{\sqrt{3}} - \sec (0)) - \frac{π}{6} + 2 \tan (\frac{π}{6}) - 2 \tan (0)$

Step 10.3.2

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

$2 (\frac{2}{\sqrt{3}} - 1 \cdot 1) - \frac{π}{6} + 2 \tan (\frac{π}{6}) - 2 \tan (0)$

Step 10.3.3

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

$2 (\frac{2}{\sqrt{3}} - 1 \cdot 1) - \frac{π}{6} + 2 \frac{\sqrt{3}}{3} - 2 \tan (0)$

Step 10.3.4

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

$2 (\frac{2}{\sqrt{3}} - 1 \cdot 1) - \frac{π}{6} + 2 \frac{\sqrt{3}}{3} - 2 \cdot 0$

Step 10.3.5

Multiply $- 1$ by $1$ .

$2 (\frac{2}{\sqrt{3}} - 1) - \frac{π}{6} + 2 \frac{\sqrt{3}}{3} - 2 \cdot 0$

Step 10.3.6

To write $2 (\frac{2}{\sqrt{3}} - 1)$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 (\frac{2}{\sqrt{3}} - 1) \cdot \frac{6}{6} - \frac{π}{6} + 2 \frac{\sqrt{3}}{3} - 2 \cdot 0$

Step 10.3.7

Combine $2 (\frac{2}{\sqrt{3}} - 1)$ and $\frac{6}{6}$ .

$\frac{2 (\frac{2}{\sqrt{3}} - 1) \cdot 6}{6} - \frac{π}{6} + 2 \frac{\sqrt{3}}{3} - 2 \cdot 0$

Step 10.3.8

Combine the numerators over the common denominator.

$\frac{2 (\frac{2}{\sqrt{3}} - 1) \cdot 6 - π}{6} + 2 \frac{\sqrt{3}}{3} - 2 \cdot 0$

Step 10.3.9

Multiply $6$ by $2$ .

$\frac{12 (\frac{2}{\sqrt{3}} - 1) - π}{6} + 2 \frac{\sqrt{3}}{3} - 2 \cdot 0$

Step 10.3.10

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

$\frac{12 (\frac{2}{\sqrt{3}} - 1) - π}{6} + \frac{2 \sqrt{3}}{3} - 2 \cdot 0$

Step 10.3.11

Multiply $- 2$ by $0$ .

$\frac{12 (\frac{2}{\sqrt{3}} - 1) - π}{6} + \frac{2 \sqrt{3}}{3} + 0$

Step 10.3.12

Add $\frac{12 (\frac{2}{\sqrt{3}} - 1) - π}{6} + \frac{2 \sqrt{3}}{3}$ and $0$ .

$\frac{12 (\frac{2}{\sqrt{3}} - 1) - π}{6} + \frac{2 \sqrt{3}}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{12 (\frac{2}{\sqrt{3}} - 1) - π}{6} + \frac{2 \sqrt{3}}{3}$

Decimal Form:

$0.94050283 \dots$