Calculus Examples

$x \cdot e^{2 x}$

Step 1

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

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

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.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Combine $x$ and $\frac{e^{2 x}}{2}$ .

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

Step 6

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

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

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $2 x$ .

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

Step 7.1.2

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 7.1.3

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

$2 \cdot 1$

Step 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

Multiply $2$ by $2$ .

$\frac{x e^{2 x}}{2} - \frac{1}{4} \int e^{u} d u$

Step 11

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

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

Step 12

Rewrite $\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C)$ as $\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{u} + C$ .

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

Step 13

Replace all occurrences of $u$ with $2 x$ .

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

Step 14

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

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