Calculus Examples

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

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

Step 1

Simplify the expression.

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Step 1.1

Rewrite

4

$4$ as

2

$2$ plus

2

$2$

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

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

Step 1.2

Rewrite

{sec (θ)}^{2 + 2}

${\sec (θ)}^{2 + 2}$ as

{sec}^{2} (θ) {sec}^{2} (θ)

$\sec^{2} (θ) \sec^{2} (θ)$ .

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

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

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

$\sec^{2} (θ)$ as

1 + {tan}^{2} (θ)

$1 + \tan^{2} (θ)$ .

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

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

Step 3

Let

u = tan (θ)

$u = \tan (θ)$ . Then

d u = {sec}^{2} (θ) d θ

$d u = \sec^{2} (θ) d θ$ , so

\frac{1}{{sec}^{2} (θ)} d u = d θ

$\frac{1}{\sec^{2} (θ)} d u = d θ$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 3.1

Let

u = tan (θ)

$u = \tan (θ)$ . Find

\frac{d u}{d θ}

$\frac{d u}{d θ}$ .

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Step 3.1.1

Differentiate

tan (θ)

$\tan (θ)$ .

\frac{d}{d θ} [tan (θ)]

$\frac{d}{d θ} [\tan (θ)]$

Step 3.1.2

The derivative of

tan (θ)

$\tan (θ)$ with respect to

θ

$θ$ is

{sec}^{2} (θ)

$\sec^{2} (θ)$ .

{sec}^{2} (θ)

$\sec^{2} (θ)$

{sec}^{2} (θ)

$\sec^{2} (θ)$

Step 3.2

Substitute the lower limit in for

θ

$θ$ in

u = tan (θ)

$u = \tan (θ)$ .

u_{lower} = tan (0)

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

Step 3.3

The exact value of

tan (0)

$\tan (0)$ is

0

$0$ .

u_{lower} = 0

$u_{lower} = 0$

Step 3.4

Substitute the upper limit in for

θ

$θ$ in

u = tan (θ)

$u = \tan (θ)$ .

u_{upper} = tan (\frac{π}{4})

$u_{upper} = \tan (\frac{π}{4})$

Step 3.5

The exact value of

tan (\frac{π}{4})

$\tan (\frac{π}{4})$ is

1

$1$ .

u_{upper} = 1

$u_{upper} = 1$

Step 3.6

The values found for

u_{lower}

$u_{lower}$ and

u_{upper}

$u_{upper}$ will be used to evaluate the definite integral.

u_{lower} = 0

$u_{lower} = 0$

u_{upper} = 1

$u_{upper} = 1$

Step 3.7

Rewrite the problem using

u

$u$ ,

d u

$d u$ , and the new limits of integration.

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

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

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

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

Step 4

Multiply

(1 + u^{2}) u^{4}

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

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

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

Step 5

Simplify.

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Step 5.1

Multiply

u^{4}

$u^{4}$ by

1

$1$ .

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

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

Step 5.2

Multiply

u^{2}

$u^{2}$ by

u^{4}

$u^{4}$ by adding the exponents.

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Step 5.2.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 5.2.2

Add

2

$2$ and

4

$4$ .

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

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

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

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

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

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

Step 6

Split the single integral into multiple integrals.

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

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

Step 7

By the Power Rule, the integral of

u^{4}

$u^{4}$ with respect to

u

$u$ is

\frac{1}{5} u^{5}

$\frac{1}{5} u^{5}$ .

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

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

Step 8

By the Power Rule, the integral of

u^{6}

$u^{6}$ with respect to

u

$u$ is

\frac{1}{7} u^{7}

$\frac{1}{7} u^{7}$ .

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

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

Step 9

Combine

\frac{1}{5} u^{5}]_{0}^{1}

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

\frac{1}{7} u^{7}]_{0}^{1}

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

\frac{1}{5} u^{5} + \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}

$\frac{1}{5} u^{5} + \frac{1}{7} u^{7}$ at

1

$1$ and at

0

$0$ .

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

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

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:

$0.3\overline{428571}$