Calculus Examples

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

Step 1

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

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

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $\tan (x)$ .

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

Step 2.1.2

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

$\sec^{2} (x)$

Step 2.2

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

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

Step 2.3

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

$u_{lower} = 0$

Step 2.4

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

$u_{upper} = \tan (\frac{π}{4})$

Step 2.5

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

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 2.7

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

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

Step 3

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{u}]_{0}^{1}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $e^{u}$ at $1$ and at $0$ .

$(e^{1}) - e^{0}$

Step 4.2

Simplify.

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Step 4.2.1

Simplify.

$e - e^{0}$

Step 4.2.2

Anything raised to $0$ is $1$ .

$e - 1 \cdot 1$

Step 4.2.3

Multiply $- 1$ by $1$ .

$e - 1$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e - 1$

Decimal Form:

$1.71828182 \dots$