Calculus Examples

$\int_{0}^{2} e^{2 x} d x$

Step 1

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 1.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \cdot 2$

Step 1.5

Multiply $2$ by $2$ .

$u_{upper} = 4$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 4$

Step 1.7

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

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

Step 2

Combine $e^{u}$ and $\frac{1}{2}$ .

$\int_{0}^{4} \frac{e^{u}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 4

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

$\frac{1}{2} e^{u}]_{0}^{4}$

Step 5

Simplify the expression.

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Step 5.1

Evaluate $e^{u}$ at $4$ and at $0$ .

$\frac{1}{2} ((e^{4}) - e^{0})$

Step 5.2

Anything raised to $0$ is $1$ .

$\frac{1}{2} (e^{4} - 1 \cdot 1)$

Step 5.3

Multiply $- 1$ by $1$ .

$\frac{1}{2} (e^{4} - 1)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2} \cdot (e^{4} - 1)$

Decimal Form:

$26.79907501 \dots$