Calculus Examples

$\int - 7 x e^{4 x - 2} d x$

Step 1

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$- 7 \int x e^{4 x - 2} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{4 x - 2}$ .

$- 7 (x (\frac{1}{4} e^{4 x - 2}) - \int \frac{1}{4} e^{4 x - 2} d x)$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{4}$ and $e^{4 x - 2}$ .

$- 7 (x \frac{e^{4 x - 2}}{4} - \int \frac{1}{4} e^{4 x - 2} d x)$

Step 3.2

Combine $x$ and $\frac{e^{4 x - 2}}{4}$ .

$- 7 (\frac{x e^{4 x - 2}}{4} - \int \frac{1}{4} e^{4 x - 2} d x)$

Step 3.3

Combine $\frac{1}{4}$ and $e^{4 x - 2}$ .

$- 7 (\frac{x e^{4 x - 2}}{4} - \int \frac{e^{4 x - 2}}{4} d x)$

Step 4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$- 7 (\frac{x e^{4 x - 2}}{4} - (\frac{1}{4} \int e^{4 x - 2} d x))$

Step 5

Let $u = 4 x - 2$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 4 x - 2$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $4 x - 2$ .

$\frac{d}{d x} [4 x - 2]$

Step 5.1.2

By the Sum Rule, the derivative of $4 x - 2$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$

Step 5.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 5.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 5.1.3.2

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

$4 \cdot 1 + \frac{d}{d x} [- 2]$

Step 5.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [- 2]$

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$4 + 0$

Step 5.1.4.2

Add $4$ and $0$ .

$4$

Step 5.2

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

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{4} \int e^{u} \frac{1}{4} d u)$

Step 6

Combine $e^{u}$ and $\frac{1}{4}$ .

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{4} \int \frac{e^{u}}{4} d u)$

Step 7

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{4} (\frac{1}{4} \int e^{u} d u))$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{4 \cdot 4} \int e^{u} d u)$

Step 8.2

Multiply $4$ by $4$ .

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{16} \int e^{u} d u)$

Step 9

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

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{16} (e^{u} + C))$

Step 10

Simplify.

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Step 10.1

Rewrite $- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{16} (e^{u} + C))$ as $- 7 (\frac{1}{4} x e^{4 x - 2} - \frac{1}{16} e^{u}) + C$ .

$- 7 (\frac{1}{4} x e^{4 x - 2} - \frac{1}{16} e^{u}) + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $\frac{1}{4}$ and $x$ .

$- 7 (\frac{x}{4} e^{4 x - 2} - \frac{1}{16} e^{u}) + C$

Step 10.2.2

Combine $\frac{x}{4}$ and $e^{4 x - 2}$ .

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{16} e^{u}) + C$

Step 10.2.3

To write $- \frac{1}{16} e^{u}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 7 (\frac{x e^{4 x - 2}}{4} - \frac{1}{16} e^{u} \cdot \frac{4}{4}) + C$

Step 10.2.4

Combine $- \frac{1}{16} e^{u}$ and $\frac{4}{4}$ .

$- 7 (\frac{x e^{4 x - 2}}{4} + \frac{- \frac{1}{16} e^{u} \cdot 4}{4}) + C$

Step 10.2.5

Combine the numerators over the common denominator.

$- 7 \frac{x e^{4 x - 2} - \frac{1}{16} e^{u} \cdot 4}{4} + C$

Step 10.2.6

Combine $e^{u}$ and $\frac{1}{16}$ .

$- 7 \frac{x e^{4 x - 2} - \frac{e^{u}}{16} \cdot 4}{4} + C$

Step 10.2.7

Multiply $4$ by $- 1$ .

$- 7 \frac{x e^{4 x - 2} - 4 \frac{e^{u}}{16}}{4} + C$

Step 10.2.8

Combine $- 4$ and $\frac{e^{u}}{16}$ .

$- 7 \frac{x e^{4 x - 2} + \frac{- 4 e^{u}}{16}}{4} + C$

Step 10.2.9

Cancel the common factor of $- 4$ and $16$ .

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Step 10.2.9.1

Factor $4$ out of $- 4 e^{u}$ .

$- 7 \frac{x e^{4 x - 2} + \frac{4 (- e^{u})}{16}}{4} + C$

Step 10.2.9.2

Cancel the common factors.

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Step 10.2.9.2.1

Factor $4$ out of $16$ .

$- 7 \frac{x e^{4 x - 2} + \frac{4 (- e^{u})}{4 (4)}}{4} + C$

Step 10.2.9.2.2

Cancel the common factor.

$- 7 \frac{x e^{4 x - 2} + \frac{4 (- e^{u})}{4 \cdot 4}}{4} + C$

Step 10.2.9.2.3

Rewrite the expression.

$- 7 \frac{x e^{4 x - 2} + \frac{- e^{u}}{4}}{4} + C$

Step 10.2.10

Move the negative in front of the fraction.

$- 7 \frac{x e^{4 x - 2} - \frac{e^{u}}{4}}{4} + C$

Step 10.2.11

To write $x e^{4 x - 2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 7 \frac{x e^{4 x - 2} \cdot \frac{4}{4} - \frac{e^{u}}{4}}{4} + C$

Step 10.2.12

Combine $x e^{4 x - 2}$ and $\frac{4}{4}$ .

$- 7 \frac{\frac{x e^{4 x - 2} \cdot 4}{4} - \frac{e^{u}}{4}}{4} + C$

Step 10.2.13

Combine the numerators over the common denominator.

$- 7 \frac{\frac{x e^{4 x - 2} \cdot 4 - e^{u}}{4}}{4} + C$

Step 10.2.14

Move $4$ to the left of $x e^{4 x - 2}$ .

$- 7 \frac{\frac{4 \cdot (x e^{4 x - 2}) - e^{u}}{4}}{4} + C$

Step 10.2.15

Rewrite $\frac{\frac{4 x e^{4 x - 2} - e^{u}}{4}}{4}$ as a product.

$- 7 (\frac{4 x e^{4 x - 2} - e^{u}}{4} \cdot \frac{1}{4}) + C$

Step 10.2.16

Multiply $\frac{4 x e^{4 x - 2} - e^{u}}{4}$ by $\frac{1}{4}$ .

$- 7 \frac{4 x e^{4 x - 2} - e^{u}}{4 \cdot 4} + C$

Step 10.2.17

Multiply $4$ by $4$ .

$- 7 \frac{4 x e^{4 x - 2} - e^{u}}{16} + C$

Step 10.2.18

Combine $- 7$ and $\frac{4 x e^{4 x - 2} - e^{u}}{16}$ .

$\frac{- 7 (4 x e^{4 x - 2} - e^{u})}{16} + C$

Step 10.2.19

Move the negative in front of the fraction.

$- \frac{7 (4 x e^{4 x - 2} - e^{u})}{16} + C$

Step 11

Replace all occurrences of $u$ with $4 x - 2$ .

$- \frac{7 (4 x e^{4 x - 2} - e^{4 x - 2})}{16} + C$

Step 12

Reorder terms.

$- \frac{7}{16} (4 x e^{4 x - 2} - e^{4 x - 2}) + C$