Calculus Examples

$\int 5 x e^{3 x} d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x e^{3 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{3 x}$ .

$5 (x (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} d x)$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$5 (x \frac{e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)$

Step 3.2

Combine $x$ and $\frac{e^{3 x}}{3}$ .

$5 (\frac{x e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)$

Step 3.3

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$5 (\frac{x e^{3 x}}{3} - \int \frac{e^{3 x}}{3} d x)$

Step 4

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$5 (\frac{x e^{3 x}}{3} - (\frac{1}{3} \int e^{3 x} d x))$

Step 5

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 5.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 5.1.3

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

$3 \cdot 1$

Step 5.1.4

Multiply $3$ by $1$ .

$3$

Step 5.2

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

$5 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int e^{u} \frac{1}{3} d u)$

Step 6

Combine $e^{u}$ and $\frac{1}{3}$ .

$5 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int \frac{e^{u}}{3} d u)$

Step 7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$5 (\frac{x e^{3 x}}{3} - \frac{1}{3} (\frac{1}{3} \int e^{u} d u))$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$5 (\frac{x e^{3 x}}{3} - \frac{1}{3 \cdot 3} \int e^{u} d u)$

Step 8.2

Multiply $3$ by $3$ .

$5 (\frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u} d u)$

Step 9

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

$5 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u} + C))$

Step 10

Rewrite $5 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u} + C))$ as $5 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u}) + C$ .

$5 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u}) + C$

Step 11

Replace all occurrences of $u$ with $3 x$ .

$5 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x}) + C$