Calculus Examples

$\int_{0}^{\frac{π}{3}} \tan^{5} (x) \sec^{4} (x) d x$

Step 1

Simplify the expression.

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Step 1.1

Rewrite $4$ as $2$ plus $2$

$\int_{0}^{\frac{π}{3}} \tan^{5} (x) {\sec (x)}^{2 + 2} d x$

Step 1.2

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

$\int_{0}^{\frac{π}{3}} \tan^{5} (x) (\sec^{2} (x) \sec^{2} (x)) d x$

Step 2

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

$\int_{0}^{\frac{π}{3}} \tan^{5} (x) ((1 + \tan^{2} (x)) \sec^{2} (x)) d x$

Step 3

Let $u = \tan (x)$ . Then $d u = \sec^{2} (x) d x$ , so $\frac{1}{\sec^{2} (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = \tan (x)$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $\tan (x)$ .

$\frac{d}{d x} [\tan (x)]$

Step 3.1.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (x)$

Step 3.2

Substitute the lower limit in for $x$ in $u = \tan (x)$ .

$u_{lower} = \tan (0)$

Step 3.3

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

$u_{lower} = 0$

Step 3.4

Substitute the upper limit in for $x$ in $u = \tan (x)$ .

$u_{upper} = \tan (\frac{π}{3})$

Step 3.5

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

$u_{upper} = \sqrt{3}$

Step 3.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \sqrt{3}$

Step 3.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{\sqrt{3}} u^{5} (1 + u^{2}) d u$

Step 4

Multiply $u^{5} (1 + u^{2})$ .

$\int_{0}^{\sqrt{3}} u^{5} \cdot 1 + u^{5} u^{2} d u$

Step 5

Simplify.

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Step 5.1

Multiply $u^{5}$ by $1$ .

$\int_{0}^{\sqrt{3}} u^{5} + u^{5} u^{2} d u$

Step 5.2

Multiply $u^{5}$ by $u^{2}$ by adding the exponents.

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Step 5.2.1

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

$\int_{0}^{\sqrt{3}} u^{5} + u^{5 + 2} d u$

Step 5.2.2

Add $5$ and $2$ .

$\int_{0}^{\sqrt{3}} u^{5} + u^{7} d u$

Step 6

Split the single integral into multiple integrals.

$\int_{0}^{\sqrt{3}} u^{5} d u + \int_{0}^{\sqrt{3}} u^{7} d u$

Step 7

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{6} u^{6}]_{0}^{\sqrt{3}} + \int_{0}^{\sqrt{3}} u^{7} d u$

Step 8

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$\frac{1}{6} u^{6}]_{0}^{\sqrt{3}} + \frac{1}{8} u^{8}]_{0}^{\sqrt{3}}$

Step 9

Combine $\frac{1}{6} u^{6}]_{0}^{\sqrt{3}}$ and $\frac{1}{8} u^{8}]_{0}^{\sqrt{3}}$ .

$\frac{1}{6} u^{6} + \frac{1}{8} u^{8}]_{0}^{\sqrt{3}}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{1}{6} u^{6} + \frac{1}{8} u^{8}$ at $\sqrt{3}$ and at $0$ .

$(\frac{1}{6} {\sqrt{3}}^{6} + \frac{1}{8} {\sqrt{3}}^{8}) - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2

Simplify.

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Step 10.2.1

Rewrite ${\sqrt{3}}^{6}$ as $3^{3}$ .

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Step 10.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{6} {(3^{\frac{1}{2}})}^{6} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{6} \cdot 3^{\frac{1}{2} \cdot 6} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.1.3

Combine $\frac{1}{2}$ and $6$ .

$\frac{1}{6} \cdot 3^{\frac{6}{2}} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.1.4

Cancel the common factor of $6$ and $2$ .

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Step 10.2.1.4.1

Factor $2$ out of $6$ .

$\frac{1}{6} \cdot 3^{\frac{2 \cdot 3}{2}} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.1.4.2

Cancel the common factors.

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Step 10.2.1.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{6} \cdot 3^{\frac{2 \cdot 3}{2 (1)}} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.1.4.2.2

Cancel the common factor.

$\frac{1}{6} \cdot 3^{\frac{2 \cdot 3}{2 \cdot 1}} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.1.4.2.3

Rewrite the expression.

$\frac{1}{6} \cdot 3^{\frac{3}{1}} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.1.4.2.4

Divide $3$ by $1$ .

$\frac{1}{6} \cdot 3^{3} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.2

Raise $3$ to the power of $3$ .

$\frac{1}{6} \cdot 27 + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.3

Combine $\frac{1}{6}$ and $27$ .

$\frac{27}{6} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.4

Cancel the common factor of $27$ and $6$ .

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Step 10.2.4.1

Factor $3$ out of $27$ .

$\frac{3 (9)}{6} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.4.2

Cancel the common factors.

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Step 10.2.4.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 9}{3 \cdot 2} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.4.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 2} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.4.2.3

Rewrite the expression.

$\frac{9}{2} + \frac{1}{8} {\sqrt{3}}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5

Rewrite ${\sqrt{3}}^{8}$ as $3^{4}$ .

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Step 10.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{9}{2} + \frac{1}{8} {(3^{\frac{1}{2}})}^{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{9}{2} + \frac{1}{8} \cdot 3^{\frac{1}{2} \cdot 8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5.3

Combine $\frac{1}{2}$ and $8$ .

$\frac{9}{2} + \frac{1}{8} \cdot 3^{\frac{8}{2}} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5.4

Cancel the common factor of $8$ and $2$ .

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Step 10.2.5.4.1

Factor $2$ out of $8$ .

$\frac{9}{2} + \frac{1}{8} \cdot 3^{\frac{2 \cdot 4}{2}} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5.4.2

Cancel the common factors.

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Step 10.2.5.4.2.1

Factor $2$ out of $2$ .

$\frac{9}{2} + \frac{1}{8} \cdot 3^{\frac{2 \cdot 4}{2 (1)}} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5.4.2.2

Cancel the common factor.

$\frac{9}{2} + \frac{1}{8} \cdot 3^{\frac{2 \cdot 4}{2 \cdot 1}} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5.4.2.3

Rewrite the expression.

$\frac{9}{2} + \frac{1}{8} \cdot 3^{\frac{4}{1}} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.5.4.2.4

Divide $4$ by $1$ .

$\frac{9}{2} + \frac{1}{8} \cdot 3^{4} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.6

Raise $3$ to the power of $4$ .

$\frac{9}{2} + \frac{1}{8} \cdot 81 - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.7

Combine $\frac{1}{8}$ and $81$ .

$\frac{9}{2} + \frac{81}{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.8

To write $\frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{9}{2} \cdot \frac{4}{4} + \frac{81}{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.9

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.9.1

Multiply $\frac{9}{2}$ by $\frac{4}{4}$ .

$\frac{9 \cdot 4}{2 \cdot 4} + \frac{81}{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.9.2

Multiply $2$ by $4$ .

$\frac{9 \cdot 4}{8} + \frac{81}{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.10

Combine the numerators over the common denominator.

$\frac{9 \cdot 4 + 81}{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.11

Simplify the numerator.

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Step 10.2.11.1

Multiply $9$ by $4$ .

$\frac{36 + 81}{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.11.2

Add $36$ and $81$ .

$\frac{117}{8} - (\frac{1}{6} \cdot 0^{6} + \frac{1}{8} \cdot 0^{8})$

Step 10.2.12

Raising $0$ to any positive power yields $0$ .

$\frac{117}{8} - (\frac{1}{6} \cdot 0 + \frac{1}{8} \cdot 0^{8})$

Step 10.2.13

Multiply $\frac{1}{6}$ by $0$ .

$\frac{117}{8} - (0 + \frac{1}{8} \cdot 0^{8})$

Step 10.2.14

Raising $0$ to any positive power yields $0$ .

$\frac{117}{8} - (0 + \frac{1}{8} \cdot 0)$

Step 10.2.15

Multiply $\frac{1}{8}$ by $0$ .

$\frac{117}{8} - (0 + 0)$

Step 10.2.16

Add $0$ and $0$ .

$\frac{117}{8} - 0$

Step 10.2.17

Multiply $- 1$ by $0$ .

$\frac{117}{8} + 0$

Step 10.2.18

Add $\frac{117}{8}$ and $0$ .

$\frac{117}{8}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{117}{8}$

Decimal Form:

$14.625$

Mixed Number Form:

$14 \frac{5}{8}$