Calculus Examples

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

Step 1

Since $4$ is constant with respect to $θ$ , move $4$ out of the integral.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $\tan (θ)$ .

$\frac{d}{d θ} [\tan (θ)]$

Step 4.1.2

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

$\sec^{2} (θ)$

Step 4.2

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

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

Step 4.3

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

$u_{lower} = 0$

Step 4.4

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

$u_{upper} = \tan (\frac{π}{4})$

Step 4.5

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

$u_{upper} = 1$

Step 4.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 4.7

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Multiply $u^{2}$ by $u^{4}$ by adding the exponents.

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Step 6.2.1

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

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

Step 6.2.2

Add $2$ and $4$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

$4 (\frac{1}{5} u^{5}]_{0}^{1} + \int_{0}^{1} u^{6} d u)$

Step 9

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

$4 (\frac{u^{5}}{5}]_{0}^{1} + \int_{0}^{1} u^{6} d u)$

Step 10

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

$4 (\frac{u^{5}}{5}]_{0}^{1} + \frac{1}{7} u^{7}]_{0}^{1})$

Step 11

Simplify.

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Step 11.1

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

$4 (\frac{u^{5}}{5} + \frac{1}{7} u^{7}]_{0}^{1})$

Step 11.2

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

$4 (\frac{u^{5}}{5} + \frac{u^{7}}{7}]_{0}^{1})$

Step 12

Substitute and simplify.

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Step 12.1

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

$4 ((\frac{1^{5}}{5} + \frac{1^{7}}{7}) - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2

Simplify.

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Step 12.2.1

One to any power is one.

$4 (\frac{1}{5} + \frac{1^{7}}{7} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.2

One to any power is one.

$4 (\frac{1}{5} + \frac{1}{7} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.3

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

$4 (\frac{1}{5} \cdot \frac{7}{7} + \frac{1}{7} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.4

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

$4 (\frac{1}{5} \cdot \frac{7}{7} + \frac{1}{7} \cdot \frac{5}{5} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.5

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

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Step 12.2.5.1

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

$4 (\frac{7}{5 \cdot 7} + \frac{1}{7} \cdot \frac{5}{5} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.5.2

Multiply $5$ by $7$ .

$4 (\frac{7}{35} + \frac{1}{7} \cdot \frac{5}{5} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.5.3

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

$4 (\frac{7}{35} + \frac{5}{7 \cdot 5} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.5.4

Multiply $7$ by $5$ .

$4 (\frac{7}{35} + \frac{5}{35} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.6

Combine the numerators over the common denominator.

$4 (\frac{7 + 5}{35} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.7

Add $7$ and $5$ .

$4 (\frac{12}{35} - (\frac{0^{5}}{5} + \frac{0^{7}}{7}))$

Step 12.2.8

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

$4 (\frac{12}{35} - (\frac{0}{5} + \frac{0^{7}}{7}))$

Step 12.2.9

Cancel the common factor of $0$ and $5$ .

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Step 12.2.9.1

Factor $5$ out of $0$ .

$4 (\frac{12}{35} - (\frac{5 (0)}{5} + \frac{0^{7}}{7}))$

Step 12.2.9.2

Cancel the common factors.

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Step 12.2.9.2.1

Factor $5$ out of $5$ .

$4 (\frac{12}{35} - (\frac{5 \cdot 0}{5 \cdot 1} + \frac{0^{7}}{7}))$

Step 12.2.9.2.2

Cancel the common factor.

$4 (\frac{12}{35} - (\frac{5 \cdot 0}{5 \cdot 1} + \frac{0^{7}}{7}))$

Step 12.2.9.2.3

Rewrite the expression.

$4 (\frac{12}{35} - (\frac{0}{1} + \frac{0^{7}}{7}))$

Step 12.2.9.2.4

Divide $0$ by $1$ .

$4 (\frac{12}{35} - (0 + \frac{0^{7}}{7}))$

Step 12.2.10

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

$4 (\frac{12}{35} - (0 + \frac{0}{7}))$

Step 12.2.11

Cancel the common factor of $0$ and $7$ .

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Step 12.2.11.1

Factor $7$ out of $0$ .

$4 (\frac{12}{35} - (0 + \frac{7 (0)}{7}))$

Step 12.2.11.2

Cancel the common factors.

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Step 12.2.11.2.1

Factor $7$ out of $7$ .

$4 (\frac{12}{35} - (0 + \frac{7 \cdot 0}{7 \cdot 1}))$

Step 12.2.11.2.2

Cancel the common factor.

$4 (\frac{12}{35} - (0 + \frac{7 \cdot 0}{7 \cdot 1}))$

Step 12.2.11.2.3

Rewrite the expression.

$4 (\frac{12}{35} - (0 + \frac{0}{1}))$

Step 12.2.11.2.4

Divide $0$ by $1$ .

$4 (\frac{12}{35} - (0 + 0))$

Step 12.2.12

Add $0$ and $0$ .

$4 (\frac{12}{35} - 0)$

Step 12.2.13

Multiply $- 1$ by $0$ .

$4 (\frac{12}{35} + 0)$

Step 12.2.14

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

$4 (\frac{12}{35})$

Step 12.2.15

Combine $4$ and $\frac{12}{35}$ .

$\frac{4 \cdot 12}{35}$

Step 12.2.16

Multiply $4$ by $12$ .

$\frac{48}{35}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{48}{35}$

Decimal Form:

$1.37142857 \dots$

Mixed Number Form:

$1 \frac{13}{35}$