Calculus Examples

$\int_{0}^{\frac{π}{4}} \tan^{3} (x) \sec^{2} (x) d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\tan (x)$ .

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

Step 1.1.2

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

$\sec^{2} (x)$

Step 1.2

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

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

Step 1.3

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

$u_{lower} = 0$

Step 1.4

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

$u_{upper} = \tan (\frac{π}{4})$

Step 1.5

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

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 1.7

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

$\int_{0}^{1} u^{3} d u$

Step 2

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{4} u^{4}]_{0}^{1}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{4} u^{4}$ at $1$ and at $0$ .

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

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$\frac{1}{4} \cdot 1 - \frac{1}{4} \cdot 0^{4}$

Step 3.2.2

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4} - \frac{1}{4} \cdot 0^{4}$

Step 3.2.3

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

$\frac{1}{4} - \frac{1}{4} \cdot 0$

Step 3.2.4

Multiply $0$ by $- 1$ .

$\frac{1}{4} + 0 (\frac{1}{4})$

Step 3.2.5

Multiply $0$ by $\frac{1}{4}$ .

$\frac{1}{4} + 0$

Step 3.2.6

Add $\frac{1}{4}$ and $0$ .

$\frac{1}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$