Calculus Examples

$\int_{0}^{\frac{π}{4}} \sec^{4} (θ) \tan^{4} (θ) d θ$

Step 1

Simplify the expression.

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Step 1.1

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

$\int_{0}^{\frac{π}{4}} {\sec (θ)}^{2 + 2} \tan^{4} (θ) d θ$

Step 1.2

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

$\int_{0}^{\frac{π}{4}} \sec^{2} (θ) \sec^{2} (θ) \tan^{4} (θ) d θ$

Step 2

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

$\int_{0}^{\frac{π}{4}} (1 + \tan^{2} (θ)) \sec^{2} (θ) \tan^{4} (θ) d θ$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $\tan (θ)$ .

$\frac{d}{d θ} [\tan (θ)]$

Step 3.1.2

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

$\sec^{2} (θ)$

Step 3.2

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

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

Step 3.3

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

$u_{lower} = 0$

Step 3.4

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

$u_{upper} = \tan (\frac{π}{4})$

Step 3.5

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$u_{upper} = 1$

Step 3.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 3.7

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

$\int_{0}^{1} (1 + u^{2}) u^{4} d u$

Step 4

Multiply $(1 + u^{2}) u^{4}$ .

$\int_{0}^{1} 1 u^{4} + u^{2} u^{4} d u$

Step 5

Simplify.

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Step 5.1

Multiply $u^{4}$ by $1$ .

$\int_{0}^{1} u^{4} + u^{2} u^{4} d u$

Step 5.2

Multiply $u^{2}$ by $u^{4}$ 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.

$\int_{0}^{1} u^{4} + u^{2 + 4} d u$

Step 5.2.2

Add $2$ and $4$ .

$\int_{0}^{1} u^{4} + u^{6} d u$

Step 6

Split the single integral into multiple integrals.

$\int_{0}^{1} u^{4} d u + \int_{0}^{1} u^{6} d u$

Step 7

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{5} u^{5}]_{0}^{1} + \int_{0}^{1} u^{6} d u$

Step 8

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$\frac{1}{5} u^{5}]_{0}^{1} + \frac{1}{7} u^{7}]_{0}^{1}$

Step 9

Combine $\frac{1}{5} u^{5}]_{0}^{1}$ and $\frac{1}{7} u^{7}]_{0}^{1}$ .

$\frac{1}{5} u^{5} + \frac{1}{7} u^{7}]_{0}^{1}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{1}{5} u^{5} + \frac{1}{7} u^{7}$ at $1$ and at $0$ .

$(\frac{1}{5} \cdot 1^{5} + \frac{1}{7} \cdot 1^{7}) - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2

Simplify.

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Step 10.2.1

One to any power is one.

$\frac{1}{5} \cdot 1 + \frac{1}{7} \cdot 1^{7} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.2

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5} + \frac{1}{7} \cdot 1^{7} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.3

One to any power is one.

$\frac{1}{5} + \frac{1}{7} \cdot 1 - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.4

Multiply $\frac{1}{7}$ by $1$ .

$\frac{1}{5} + \frac{1}{7} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.5

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{1}{5} \cdot \frac{7}{7} + \frac{1}{7} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.6

To write $\frac{1}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} \cdot \frac{7}{7} + \frac{1}{7} \cdot \frac{5}{5} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.7

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.7.1

Multiply $\frac{1}{5}$ by $\frac{7}{7}$ .

$\frac{7}{5 \cdot 7} + \frac{1}{7} \cdot \frac{5}{5} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.7.2

Multiply $5$ by $7$ .

$\frac{7}{35} + \frac{1}{7} \cdot \frac{5}{5} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.7.3

Multiply $\frac{1}{7}$ by $\frac{5}{5}$ .

$\frac{7}{35} + \frac{5}{7 \cdot 5} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.7.4

Multiply $7$ by $5$ .

$\frac{7}{35} + \frac{5}{35} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.8

Combine the numerators over the common denominator.

$\frac{7 + 5}{35} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.9

Add $7$ and $5$ .

$\frac{12}{35} - (\frac{1}{5} \cdot 0^{5} + \frac{1}{7} \cdot 0^{7})$

Step 10.2.10

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

$\frac{12}{35} - (\frac{1}{5} \cdot 0 + \frac{1}{7} \cdot 0^{7})$

Step 10.2.11

Multiply $\frac{1}{5}$ by $0$ .

$\frac{12}{35} - (0 + \frac{1}{7} \cdot 0^{7})$

Step 10.2.12

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

$\frac{12}{35} - (0 + \frac{1}{7} \cdot 0)$

Step 10.2.13

Multiply $\frac{1}{7}$ by $0$ .

$\frac{12}{35} - (0 + 0)$

Step 10.2.14

Add $0$ and $0$ .

$\frac{12}{35} - 0$

Step 10.2.15

Multiply $- 1$ by $0$ .

$\frac{12}{35} + 0$

Step 10.2.16

Add $\frac{12}{35}$ and $0$ .

$\frac{12}{35}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{35}$

Decimal Form:

$0.3\overline{428571}$