Calculus Examples

$x e^{\frac{x}{2}}$

Step 1

Write $x e^{\frac{x}{2}}$ as a function.

$f (x) = x e^{\frac{x}{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x e^{\frac{x}{2}} d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{\frac{x}{2}}$ .

$x (2 e^{\frac{1}{2} x}) - \int 2 e^{\frac{1}{2} x} d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$x (2 e^{\frac{1}{2} x}) - (2 \int e^{\frac{1}{2} x} d x)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $x$ .

$x (2 e^{\frac{x}{2}}) - (2 \int e^{\frac{1}{2} x} d x)$

Step 6.2

Combine $\frac{1}{2}$ and $x$ .

$x (2 e^{\frac{x}{2}}) - (2 \int e^{\frac{x}{2}} d x)$

Step 6.3

Multiply $2$ by $- 1$ .

$x (2 e^{\frac{x}{2}}) - 2 \int e^{\frac{x}{2}} d x$

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $\frac{x}{2}$ .

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

Step 7.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{1}{2} \frac{d}{d x} [x]$

Step 7.1.3

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

$\frac{1}{2} \cdot 1$

Step 7.1.4

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 7.2

Rewrite the problem using $u$ and $d u$ .

$x (2 e^{\frac{x}{2}}) - 2 \int e^{u} \frac{1}{\frac{1}{2}} d u$

Step 8

Simplify.

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Step 8.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$x (2 e^{\frac{x}{2}}) - 2 \int e^{u} (1 \cdot 2) d u$

Step 8.2

Multiply $2$ by $1$ .

$x (2 e^{\frac{x}{2}}) - 2 \int e^{u} \cdot 2 d u$

Step 8.3

Move $2$ to the left of $e^{u}$ .

$x (2 e^{\frac{x}{2}}) - 2 \int 2 e^{u} d u$

Step 9

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$x (2 e^{\frac{x}{2}}) - 2 (2 \int e^{u} d u)$

Step 10

Multiply $2$ by $- 2$ .

$x (2 e^{\frac{x}{2}}) - 4 \int e^{u} d u$

Step 11

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

$x (2 e^{\frac{x}{2}}) - 4 (e^{u} + C)$

Step 12

Rewrite $x (2 e^{\frac{x}{2}}) - 4 (e^{u} + C)$ as $2 x e^{\frac{x}{2}} - 4 e^{u} + C$ .

$2 x e^{\frac{x}{2}} - 4 e^{u} + C$

Step 13

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$2 x e^{\frac{x}{2}} - 4 e^{\frac{x}{2}} + C$

Step 14

Reorder terms.

$2 x e^{\frac{1}{2} x} - 4 e^{\frac{1}{2} x} + C$

Step 15

The answer is the antiderivative of the function $f (x) = x e^{\frac{x}{2}}$ .

$F (x) =$ $2 x e^{\frac{1}{2} x} - 4 e^{\frac{1}{2} x} + C$