Calculus Examples

$\int 3 x e^{- 9 x} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x e^{- 9 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 9 x}$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2

Multiply $\int \frac{e^{- 9 x}}{9} d x$ by $1$ .

$3 (- \frac{x e^{- 9 x}}{9} + \int \frac{e^{- 9 x}}{9} d x)$

Step 6

Since $\frac{1}{9}$ is constant with respect to $x$ , move $\frac{1}{9}$ out of the integral.

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

Step 7

Let $u = - 9 x$ . Then $d u = - 9 d x$ , so $- \frac{1}{9} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1

Let $u = - 9 x$ . Find $\frac{d u}{d x}$ .

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Step 7.1.1

Differentiate $- 9 x$ .

$\frac{d}{d x} [- 9 x]$

Step 7.1.2

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$- 9 \frac{d}{d x} [x]$

Step 7.1.3

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

$- 9 \cdot 1$

Step 7.1.4

Multiply $- 9$ by $1$ .

$- 9$

Step 7.2

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

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

Step 8

Simplify.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 9

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

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

Step 10

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

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

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{9}$ by $\frac{1}{9}$ .

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

Step 11.2

Multiply $9$ by $9$ .

$3 (- \frac{x e^{- 9 x}}{9} - \frac{1}{81} \int e^{u} d u)$

Step 12

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

$3 (- \frac{x e^{- 9 x}}{9} - \frac{1}{81} (e^{u} + C))$

Step 13

Simplify.

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Step 13.1

Rewrite $3 (- \frac{x e^{- 9 x}}{9} - \frac{1}{81} (e^{u} + C))$ as $3 (- \frac{1}{9} x e^{- 9 x} - \frac{1}{81} e^{u}) + C$ .

$3 (- \frac{1}{9} x e^{- 9 x} - \frac{1}{81} e^{u}) + C$

Step 13.2

Simplify.

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Step 13.2.1

Combine $x$ and $\frac{1}{9}$ .

$3 (- \frac{x}{9} e^{- 9 x} - \frac{1}{81} e^{u}) + C$

Step 13.2.2

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

$3 (- \frac{e^{- 9 x} x}{9} - \frac{1}{81} e^{u}) + C$

Step 13.2.3

Combine $e^{u}$ and $\frac{1}{81}$ .

$3 (- \frac{e^{- 9 x} x}{9} - \frac{e^{u}}{81}) + C$

Step 14

Replace all occurrences of $u$ with $- 9 x$ .

$3 (- \frac{e^{- 9 x} x}{9} - \frac{e^{- 9 x}}{81}) + C$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$3 (- \frac{e^{- 9 x} x}{9}) + 3 (- \frac{e^{- 9 x}}{81}) + C$

Step 15.2

Cancel the common factor of $3$ .

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Step 15.2.1

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

$3 \frac{- e^{- 9 x} x}{9} + 3 (- \frac{e^{- 9 x}}{81}) + C$

Step 15.2.2

Factor $3$ out of $9$ .

$3 \frac{- e^{- 9 x} x}{3 (3)} + 3 (- \frac{e^{- 9 x}}{81}) + C$

Step 15.2.3

Cancel the common factor.

$3 \frac{- e^{- 9 x} x}{3 \cdot 3} + 3 (- \frac{e^{- 9 x}}{81}) + C$

Step 15.2.4

Rewrite the expression.

$\frac{- e^{- 9 x} x}{3} + 3 (- \frac{e^{- 9 x}}{81}) + C$

Step 15.3

Cancel the common factor of $3$ .

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Step 15.3.1

Move the leading negative in $- \frac{e^{- 9 x}}{81}$ into the numerator.

$\frac{- e^{- 9 x} x}{3} + 3 \frac{- e^{- 9 x}}{81} + C$

Step 15.3.2

Factor $3$ out of $81$ .

$\frac{- e^{- 9 x} x}{3} + 3 \frac{- e^{- 9 x}}{3 (27)} + C$

Step 15.3.3

Cancel the common factor.

$\frac{- e^{- 9 x} x}{3} + 3 \frac{- e^{- 9 x}}{3 \cdot 27} + C$

Step 15.3.4

Rewrite the expression.

$\frac{- e^{- 9 x} x}{3} + \frac{- e^{- 9 x}}{27} + C$

Step 15.4

Simplify each term.

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Step 15.4.1

Move the negative in front of the fraction.

$- \frac{e^{- 9 x} x}{3} + \frac{- e^{- 9 x}}{27} + C$

Step 15.4.2

Move the negative in front of the fraction.

$- \frac{e^{- 9 x} x}{3} - \frac{e^{- 9 x}}{27} + C$

Step 15.5

Reorder factors in $- \frac{e^{- 9 x} x}{3} - \frac{e^{- 9 x}}{27}$ .

$- \frac{x e^{- 9 x}}{3} - \frac{e^{- 9 x}}{27} + C$

Step 16

Reorder terms.

$- \frac{1}{3} x e^{- 9 x} - \frac{1}{27} e^{- 9 x} + C$