Calculus Examples

$\int 2 x e^{5 x} d x$

Step 1

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

$2 \int x e^{5 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{5 x}$ .

$2 (x (\frac{1}{5} e^{5 x}) - \int \frac{1}{5} e^{5 x} d x)$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$2 (x \frac{e^{5 x}}{5} - \int \frac{1}{5} e^{5 x} d x)$

Step 3.2

Combine $x$ and $\frac{e^{5 x}}{5}$ .

$2 (\frac{x e^{5 x}}{5} - \int \frac{1}{5} e^{5 x} d x)$

Step 3.3

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$2 (\frac{x e^{5 x}}{5} - \int \frac{e^{5 x}}{5} d x)$

Step 4

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$2 (\frac{x e^{5 x}}{5} - (\frac{1}{5} \int e^{5 x} d x))$

Step 5

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 5.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 5.1.3

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

$5 \cdot 1$

Step 5.1.4

Multiply $5$ by $1$ .

$5$

Step 5.2

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

$2 (\frac{x e^{5 x}}{5} - \frac{1}{5} \int e^{u} \frac{1}{5} d u)$

Step 6

Combine $e^{u}$ and $\frac{1}{5}$ .

$2 (\frac{x e^{5 x}}{5} - \frac{1}{5} \int \frac{e^{u}}{5} d u)$

Step 7

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$2 (\frac{x e^{5 x}}{5} - \frac{1}{5} (\frac{1}{5} \int e^{u} d u))$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$2 (\frac{x e^{5 x}}{5} - \frac{1}{5 \cdot 5} \int e^{u} d u)$

Step 8.2

Multiply $5$ by $5$ .

$2 (\frac{x e^{5 x}}{5} - \frac{1}{25} \int e^{u} d u)$

Step 9

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

$2 (\frac{x e^{5 x}}{5} - \frac{1}{25} (e^{u} + C))$

Step 10

Rewrite $2 (\frac{x e^{5 x}}{5} - \frac{1}{25} (e^{u} + C))$ as $2 (\frac{1}{5} x e^{5 x} - \frac{1}{25} e^{u}) + C$ .

$2 (\frac{1}{5} x e^{5 x} - \frac{1}{25} e^{u}) + C$

Step 11

Replace all occurrences of $u$ with $5 x$ .

$2 (\frac{1}{5} x e^{5 x} - \frac{1}{25} e^{5 x}) + C$