Calculus Examples

$\int \frac{3}{2} \cdot (x e^{- 6 x}) d x$

Step 1

Simplify.

Tap for more steps...

Step 1.1

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

$\int \frac{x \cdot 3}{2} \cdot e^{- 6 x} d x$

Step 1.2

Combine $\frac{x \cdot 3}{2}$ and $e^{- 6 x}$ .

$\int \frac{x \cdot 3 e^{- 6 x}}{2} d x$

Step 1.3

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

$\int \frac{3 x e^{- 6 x}}{2} d x$

Step 2

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

$\frac{3}{2} \int x e^{- 6 x} d x$

Step 3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 6 x}$ .

$\frac{3}{2} (x (- \frac{1}{6} e^{- 6 x}) - \int - \frac{1}{6} e^{- 6 x} d x)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Combine $e^{- 6 x}$ and $\frac{1}{6}$ .

$\frac{3}{2} (x (- \frac{e^{- 6 x}}{6}) - \int - \frac{1}{6} e^{- 6 x} d x)$

Step 4.2

Combine $x$ and $\frac{e^{- 6 x}}{6}$ .

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} - \int - \frac{1}{6} e^{- 6 x} d x)$

Step 4.3

Combine $e^{- 6 x}$ and $\frac{1}{6}$ .

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} - \int - \frac{e^{- 6 x}}{6} d x)$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} - - \int \frac{e^{- 6 x}}{6} d x)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + 1 \int \frac{e^{- 6 x}}{6} d x)$

Step 6.2

Multiply $\int \frac{e^{- 6 x}}{6} d x$ by $1$ .

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + \int \frac{e^{- 6 x}}{6} d x)$

Step 7

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + \frac{1}{6} \int e^{- 6 x} d x)$

Step 8

Let $u = - 6 x$ . Then $d u = - 6 d x$ , so $- \frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 8.1

Let $u = - 6 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 8.1.1

Differentiate $- 6 x$ .

$\frac{d}{d x} [- 6 x]$

Step 8.1.2

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$- 6 \frac{d}{d x} [x]$

Step 8.1.3

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

$- 6 \cdot 1$

Step 8.1.4

Multiply $- 6$ by $1$ .

$- 6$

Step 8.2

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

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + \frac{1}{6} \int e^{u} \frac{1}{- 6} d u)$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Move the negative in front of the fraction.

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + \frac{1}{6} \int e^{u} (- \frac{1}{6}) d u)$

Step 9.2

Combine $e^{u}$ and $\frac{1}{6}$ .

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + \frac{1}{6} \int - \frac{e^{u}}{6} d u)$

Step 10

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + \frac{1}{6} (- \int \frac{e^{u}}{6} d u))$

Step 11

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} + \frac{1}{6} (- (\frac{1}{6} \int e^{u} d u)))$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Multiply $\frac{1}{6}$ by $\frac{1}{6}$ .

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} - \frac{1}{6 \cdot 6} \int e^{u} d u)$

Step 12.2

Multiply $6$ by $6$ .

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} - \frac{1}{36} \int e^{u} d u)$

Step 13

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

$\frac{3}{2} (- \frac{x e^{- 6 x}}{6} - \frac{1}{36} (e^{u} + C))$

Step 14

Simplify.

Tap for more steps...

Step 14.1

Rewrite $\frac{3}{2} (- \frac{x e^{- 6 x}}{6} - \frac{1}{36} (e^{u} + C))$ as $\frac{3}{2} (- \frac{1}{6} x e^{- 6 x} - \frac{1}{36} e^{u}) + C$ .

$\frac{3}{2} (- \frac{1}{6} x e^{- 6 x} - \frac{1}{36} e^{u}) + C$

Step 14.2

Simplify.

Tap for more steps...

Step 14.2.1

Combine $x$ and $\frac{1}{6}$ .

$\frac{3}{2} (- \frac{x}{6} e^{- 6 x} - \frac{1}{36} e^{u}) + C$

Step 14.2.2

Combine $e^{- 6 x}$ and $\frac{x}{6}$ .

$\frac{3}{2} (- \frac{e^{- 6 x} x}{6} - \frac{1}{36} e^{u}) + C$

Step 14.2.3

Combine $e^{u}$ and $\frac{1}{36}$ .

$\frac{3}{2} (- \frac{e^{- 6 x} x}{6} - \frac{e^{u}}{36}) + C$

Step 15

Replace all occurrences of $u$ with $- 6 x$ .

$\frac{3}{2} (- \frac{e^{- 6 x} x}{6} - \frac{e^{- 6 x}}{36}) + C$

Step 16

Simplify.

Tap for more steps...

Step 16.1

Apply the distributive property.

$\frac{3}{2} (- \frac{e^{- 6 x} x}{6}) + \frac{3}{2} (- \frac{e^{- 6 x}}{36}) + C$

Step 16.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 16.2.1

Move the leading negative in $- \frac{e^{- 6 x} x}{6}$ into the numerator.

$\frac{3}{2} \cdot \frac{- e^{- 6 x} x}{6} + \frac{3}{2} (- \frac{e^{- 6 x}}{36}) + C$

Step 16.2.2

Factor $3$ out of $6$ .

$\frac{3}{2} \cdot \frac{- e^{- 6 x} x}{3 (2)} + \frac{3}{2} (- \frac{e^{- 6 x}}{36}) + C$

Step 16.2.3

Cancel the common factor.

$\frac{3}{2} \cdot \frac{- e^{- 6 x} x}{3 \cdot 2} + \frac{3}{2} (- \frac{e^{- 6 x}}{36}) + C$

Step 16.2.4

Rewrite the expression.

$\frac{1}{2} \cdot \frac{- e^{- 6 x} x}{2} + \frac{3}{2} (- \frac{e^{- 6 x}}{36}) + C$

Step 16.3

Multiply $\frac{1}{2}$ by $\frac{- e^{- 6 x} x}{2}$ .

$\frac{- e^{- 6 x} x}{2 \cdot 2} + \frac{3}{2} (- \frac{e^{- 6 x}}{36}) + C$

Step 16.4

Multiply $2$ by $2$ .

$\frac{- e^{- 6 x} x}{4} + \frac{3}{2} (- \frac{e^{- 6 x}}{36}) + C$

Step 16.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 16.5.1

Move the leading negative in $- \frac{e^{- 6 x}}{36}$ into the numerator.

$\frac{- e^{- 6 x} x}{4} + \frac{3}{2} \cdot \frac{- e^{- 6 x}}{36} + C$

Step 16.5.2

Factor $3$ out of $36$ .

$\frac{- e^{- 6 x} x}{4} + \frac{3}{2} \cdot \frac{- e^{- 6 x}}{3 (12)} + C$

Step 16.5.3

Cancel the common factor.

$\frac{- e^{- 6 x} x}{4} + \frac{3}{2} \cdot \frac{- e^{- 6 x}}{3 \cdot 12} + C$

Step 16.5.4

Rewrite the expression.

$\frac{- e^{- 6 x} x}{4} + \frac{1}{2} \cdot \frac{- e^{- 6 x}}{12} + C$

Step 16.6

Multiply $\frac{1}{2}$ by $\frac{- e^{- 6 x}}{12}$ .

$\frac{- e^{- 6 x} x}{4} + \frac{- e^{- 6 x}}{2 \cdot 12} + C$

Step 16.7

Multiply $2$ by $12$ .

$\frac{- e^{- 6 x} x}{4} + \frac{- e^{- 6 x}}{24} + C$

Step 16.8

Simplify each term.

Tap for more steps...

Step 16.8.1

Move the negative in front of the fraction.

$- \frac{e^{- 6 x} x}{4} + \frac{- e^{- 6 x}}{24} + C$

Step 16.8.2

Move the negative in front of the fraction.

$- \frac{e^{- 6 x} x}{4} - \frac{e^{- 6 x}}{24} + C$

Step 16.9

Reorder factors in $- \frac{e^{- 6 x} x}{4} - \frac{e^{- 6 x}}{24}$ .

$- \frac{x e^{- 6 x}}{4} - \frac{e^{- 6 x}}{24} + C$

Step 17

Reorder terms.

$- \frac{1}{4} x e^{- 6 x} - \frac{1}{24} e^{- 6 x} + C$