Calculus Examples

$6 x e^{3 x}$

Step 1

Write $6 x e^{3 x}$ as a function.

$f (x) = 6 x e^{3 x}$

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.

$F (x) = \int 6 x e^{3 x} d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x e^{3 x} d x$

Step 5

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

$6 (x (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} d x)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$6 (x \frac{e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)$

Step 6.2

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

$6 (\frac{x e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x)$

Step 6.3

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

$6 (\frac{x e^{3 x}}{3} - \int \frac{e^{3 x}}{3} d x)$

Step 7

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

$6 (\frac{x e^{3 x}}{3} - (\frac{1}{3} \int e^{3 x} d x))$

Step 8

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

Tap for more steps...

Step 8.1

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

Tap for more steps...

Step 8.1.1

Differentiate $3 x$ .

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

Step 8.1.2

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 8.1.3

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 8.1.4

Multiply $3$ by $1$ .

$3$

Step 8.2

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

$6 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int e^{u} \frac{1}{3} d u)$

Step 9

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

$6 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int \frac{e^{u}}{3} d u)$

Step 10

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

$6 (\frac{x e^{3 x}}{3} - \frac{1}{3} (\frac{1}{3} \int e^{u} d u))$

Step 11

Simplify.

Tap for more steps...

Step 11.1

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

$6 (\frac{x e^{3 x}}{3} - \frac{1}{3 \cdot 3} \int e^{u} d u)$

Step 11.2

Multiply $3$ by $3$ .

$6 (\frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u} d u)$

Step 12

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

$6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u} + C))$

Step 13

Rewrite $6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u} + C))$ as $6 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u}) + C$ .

$6 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u}) + C$

Step 14

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

$6 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x}) + C$

Step 15

Simplify.

Tap for more steps...

Step 15.1

Simplify each term.

Tap for more steps...

Step 15.1.1

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

$6 (\frac{x}{3} e^{3 x} - \frac{1}{9} e^{3 x}) + C$

Step 15.1.2

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

$6 (\frac{x e^{3 x}}{3} - \frac{1}{9} e^{3 x}) + C$

Step 15.1.3

Combine $e^{3 x}$ and $\frac{1}{9}$ .

$6 (\frac{x e^{3 x}}{3} - \frac{e^{3 x}}{9}) + C$

Step 15.2

Apply the distributive property.

$6 \frac{x e^{3 x}}{3} + 6 (- \frac{e^{3 x}}{9}) + C$

Step 15.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.3.1

Factor $3$ out of $6$ .

$3 (2) \frac{x e^{3 x}}{3} + 6 (- \frac{e^{3 x}}{9}) + C$

Step 15.3.2

Cancel the common factor.

$3 \cdot 2 \frac{x e^{3 x}}{3} + 6 (- \frac{e^{3 x}}{9}) + C$

Step 15.3.3

Rewrite the expression.

$2 (x e^{3 x}) + 6 (- \frac{e^{3 x}}{9}) + C$

Step 15.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.4.1

Move the leading negative in $- \frac{e^{3 x}}{9}$ into the numerator.

$2 (x e^{3 x}) + 6 \frac{- e^{3 x}}{9} + C$

Step 15.4.2

Factor $3$ out of $6$ .

$2 (x e^{3 x}) + 3 (2) \frac{- e^{3 x}}{9} + C$

Step 15.4.3

Factor $3$ out of $9$ .

$2 (x e^{3 x}) + 3 \cdot 2 \frac{- e^{3 x}}{3 \cdot 3} + C$

Step 15.4.4

Cancel the common factor.

$2 (x e^{3 x}) + 3 \cdot 2 \frac{- e^{3 x}}{3 \cdot 3} + C$

Step 15.4.5

Rewrite the expression.

$2 (x e^{3 x}) + 2 \frac{- e^{3 x}}{3} + C$

Step 15.5

Combine $2$ and $\frac{- e^{3 x}}{3}$ .

$2 (x e^{3 x}) + \frac{2 (- e^{3 x})}{3} + C$

Step 15.6

Multiply $- 1$ by $2$ .

$2 (x e^{3 x}) + \frac{- 2 e^{3 x}}{3} + C$

Step 15.7

Move the negative in front of the fraction.

$2 x e^{3 x} - \frac{2 e^{3 x}}{3} + C$

Step 15.8

To write $2 x e^{3 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 x e^{3 x} \cdot \frac{3}{3} - \frac{2 e^{3 x}}{3} + C$

Step 15.9

Combine $2 x e^{3 x}$ and $\frac{3}{3}$ .

$\frac{2 x e^{3 x} \cdot 3}{3} - \frac{2 e^{3 x}}{3} + C$

Step 15.10

Combine the numerators over the common denominator.

$\frac{2 x e^{3 x} \cdot 3 - 2 e^{3 x}}{3} + C$

Step 15.11

Simplify the numerator.

Tap for more steps...

Step 15.11.1

Factor $2 e^{3 x}$ out of $2 x e^{3 x} \cdot 3 - 2 e^{3 x}$ .

Tap for more steps...

Step 15.11.1.1

Factor $2 e^{3 x}$ out of $2 x e^{3 x} \cdot 3$ .

$\frac{2 e^{3 x} (x \cdot 3) - 2 e^{3 x}}{3} + C$

Step 15.11.1.2

Factor $2 e^{3 x}$ out of $- 2 e^{3 x}$ .

$\frac{2 e^{3 x} (x \cdot 3) + 2 e^{3 x} (- 1)}{3} + C$

Step 15.11.1.3

Factor $2 e^{3 x}$ out of $2 e^{3 x} (x \cdot 3) + 2 e^{3 x} (- 1)$ .

$\frac{2 e^{3 x} (x \cdot 3 - 1)}{3} + C$

Step 15.11.2

Move $3$ to the left of $x$ .

$\frac{2 e^{3 x} (3 x - 1)}{3} + C$

$\frac{2}{3} e^{3 x} (3 x - 1) + C$

Step 16

The answer is the antiderivative of the function $f (x) = 6 x e^{3 x}$ .

$F (x) =$ $\frac{2}{3} e^{3 x} (3 x - 1) + C$