Calculus Examples

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

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 2

Simplify the expression.

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Step 2.1

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

$5 \int_{0}^{\frac{π}{4}} {\sec (x)}^{2 + 2} d x$

Step 2.2

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

$5 \int_{0}^{\frac{π}{4}} \sec^{2} (x) \sec^{2} (x) d x$

Step 3

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

$5 \int_{0}^{\frac{π}{4}} (1 + \tan^{2} (x)) \sec^{2} (x) d x$

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $\tan (x)$ .

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

Step 4.1.2

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

$\sec^{2} (x)$

Step 4.2

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

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

Step 4.3

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

$u_{lower} = 0$

Step 4.4

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

$u_{upper} = \tan (\frac{π}{4})$

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.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

$5 (u]_{0}^{1} + \int_{0}^{1} u^{2} d u)$

Step 7

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$5 (u]_{0}^{1} + \frac{1}{3} u^{3}]_{0}^{1})$

Step 8

Simplify.

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Step 8.1

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

$5 (u + \frac{1}{3} u^{3}]_{0}^{1})$

Step 8.2

Combine $\frac{1}{3}$ and $u^{3}$ .

$5 (u + \frac{u^{3}}{3}]_{0}^{1})$

Step 9

Substitute and simplify.

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Step 9.1

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

$5 ((1 + \frac{1^{3}}{3}) - (0 + \frac{0^{3}}{3}))$

Step 9.2

Simplify.

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Step 9.2.1

One to any power is one.

$5 (1 + \frac{1}{3} - (0 + \frac{0^{3}}{3}))$

Step 9.2.2

Write $1$ as a fraction with a common denominator.

$5 (\frac{3}{3} + \frac{1}{3} - (0 + \frac{0^{3}}{3}))$

Step 9.2.3

Combine the numerators over the common denominator.

$5 (\frac{3 + 1}{3} - (0 + \frac{0^{3}}{3}))$

Step 9.2.4

Add $3$ and $1$ .

$5 (\frac{4}{3} - (0 + \frac{0^{3}}{3}))$

Step 9.2.5

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

$5 (\frac{4}{3} - (0 + \frac{0}{3}))$

Step 9.2.6

Cancel the common factor of $0$ and $3$ .

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Step 9.2.6.1

Factor $3$ out of $0$ .

$5 (\frac{4}{3} - (0 + \frac{3 (0)}{3}))$

Step 9.2.6.2

Cancel the common factors.

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Step 9.2.6.2.1

Factor $3$ out of $3$ .

$5 (\frac{4}{3} - (0 + \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.6.2.2

Cancel the common factor.

$5 (\frac{4}{3} - (0 + \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.6.2.3

Rewrite the expression.

$5 (\frac{4}{3} - (0 + \frac{0}{1}))$

Step 9.2.6.2.4

Divide $0$ by $1$ .

$5 (\frac{4}{3} - (0 + 0))$

Step 9.2.7

Add $0$ and $0$ .

$5 (\frac{4}{3} - 0)$

Step 9.2.8

Multiply $- 1$ by $0$ .

$5 (\frac{4}{3} + 0)$

Step 9.2.9

Add $\frac{4}{3}$ and $0$ .

$5 (\frac{4}{3})$

Step 9.2.10

Combine $5$ and $\frac{4}{3}$ .

$\frac{5 \cdot 4}{3}$

Step 9.2.11

Multiply $5$ by $4$ .

$\frac{20}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{20}{3}$

Decimal Form:

$6.\overline{6}$

Mixed Number Form:

$6 \frac{2}{3}$