Calculus Examples

$\int x^{2} e^{8 x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{8 x}$ .

$x^{2} (\frac{1}{8} e^{8 x}) - \int \frac{1}{8} e^{8 x} (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{8}$ and $e^{8 x}$ .

$x^{2} \frac{e^{8 x}}{8} - \int \frac{1}{8} e^{8 x} (2 x) d x$

Step 2.2

Combine $x^{2}$ and $\frac{e^{8 x}}{8}$ .

$\frac{x^{2} e^{8 x}}{8} - \int \frac{1}{8} e^{8 x} (2 x) d x$

Step 3

Since $\frac{1}{8} \cdot 2$ is constant with respect to $x$ , move $\frac{1}{8} \cdot 2$ out of the integral.

$\frac{x^{2} e^{8 x}}{8} - (\frac{1}{8} \cdot 2 \int e^{8 x} (x) d x)$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{8}$ and $2$ .

$\frac{x^{2} e^{8 x}}{8} - (\frac{2}{8} \int e^{8 x} (x) d x)$

Step 4.2

Cancel the common factor of $2$ and $8$ .

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Step 4.2.1

Factor $2$ out of $2$ .

$\frac{x^{2} e^{8 x}}{8} - (\frac{2 (1)}{8} \int e^{8 x} (x) d x)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $8$ .

$\frac{x^{2} e^{8 x}}{8} - (\frac{2 \cdot 1}{2 \cdot 4} \int e^{8 x} (x) d x)$

Step 4.2.2.2

Cancel the common factor.

$\frac{x^{2} e^{8 x}}{8} - (\frac{2 \cdot 1}{2 \cdot 4} \int e^{8 x} (x) d x)$

Step 4.2.2.3

Rewrite the expression.

$\frac{x^{2} e^{8 x}}{8} - (\frac{1}{4} \int e^{8 x} (x) d x)$

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} \int e^{8 x} (x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{8 x}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (x (\frac{1}{8} e^{8 x}) - \int \frac{1}{8} e^{8 x} d x)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{8}$ and $e^{8 x}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (x \frac{e^{8 x}}{8} - \int \frac{1}{8} e^{8 x} d x)$

Step 6.2

Combine $x$ and $\frac{e^{8 x}}{8}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \int \frac{1}{8} e^{8 x} d x)$

Step 6.3

Combine $\frac{1}{8}$ and $e^{8 x}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \int \frac{e^{8 x}}{8} d x)$

Step 7

Since $\frac{1}{8}$ is constant with respect to $x$ , move $\frac{1}{8}$ out of the integral.

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - (\frac{1}{8} \int e^{8 x} d x))$

Step 8

Let $u = 8 x$ . Then $d u = 8 d x$ , so $\frac{1}{8} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = 8 x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $8 x$ .

$\frac{d}{d x} [8 x]$

Step 8.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x]$

Step 8.1.3

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

$8 \cdot 1$

Step 8.1.4

Multiply $8$ by $1$ .

$8$

Step 8.2

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

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{8} \int e^{u} \frac{1}{8} d u)$

Step 9

Combine $e^{u}$ and $\frac{1}{8}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{8} \int \frac{e^{u}}{8} d u)$

Step 10

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{8} (\frac{1}{8} \int e^{u} d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{8}$ by $\frac{1}{8}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{8 \cdot 8} \int e^{u} d u)$

Step 11.2

Multiply $8$ by $8$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{64} \int e^{u} d u)$

Step 12

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

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{64} (e^{u} + C))$

Step 13

Simplify.

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Step 13.1

Rewrite $\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{64} (e^{u} + C))$ as $\frac{1}{8} x^{2} e^{8 x} - \frac{1}{4} (\frac{1}{8} x e^{8 x} - \frac{1}{64} e^{u}) + C$ .

$\frac{1}{8} x^{2} e^{8 x} - \frac{1}{4} (\frac{1}{8} x e^{8 x} - \frac{1}{64} e^{u}) + C$

Step 13.2

Simplify.

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Step 13.2.1

Combine $\frac{1}{8}$ and $x^{2}$ .

$\frac{x^{2}}{8} e^{8 x} - \frac{1}{4} (\frac{1}{8} x e^{8 x} - \frac{1}{64} e^{u}) + C$

Step 13.2.2

Combine $\frac{x^{2}}{8}$ and $e^{8 x}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{1}{8} x e^{8 x} - \frac{1}{64} e^{u}) + C$

Step 13.2.3

Combine $\frac{1}{8}$ and $x$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x}{8} e^{8 x} - \frac{1}{64} e^{u}) + C$

Step 13.2.4

Combine $\frac{x}{8}$ and $e^{8 x}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{1}{64} e^{u}) + C$

Step 13.2.5

Combine $e^{u}$ and $\frac{1}{64}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64}) + C$

Step 13.2.6

To write $- \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{x^{2} e^{8 x}}{8} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64}) \cdot \frac{8}{8} + C$

Step 13.2.7

Combine $- \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})$ and $\frac{8}{8}$ .

$\frac{x^{2} e^{8 x}}{8} + \frac{- \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64}) \cdot 8}{8} + C$

Step 13.2.8

Combine the numerators over the common denominator.

$\frac{x^{2} e^{8 x} - \frac{1}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64}) \cdot 8}{8} + C$

Step 13.2.9

Multiply $8$ by $- 1$ .

$\frac{x^{2} e^{8 x} - 8 (\frac{1}{4}) (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})}{8} + C$

Step 13.2.10

Combine $- 8$ and $\frac{1}{4}$ .

$\frac{x^{2} e^{8 x} + \frac{- 8}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})}{8} + C$

Step 13.2.11

Cancel the common factor of $- 8$ and $4$ .

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Step 13.2.11.1

Factor $4$ out of $- 8$ .

$\frac{x^{2} e^{8 x} + \frac{4 \cdot - 2}{4} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})}{8} + C$

Step 13.2.11.2

Cancel the common factors.

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Step 13.2.11.2.1

Factor $4$ out of $4$ .

$\frac{x^{2} e^{8 x} + \frac{4 \cdot - 2}{4 (1)} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})}{8} + C$

Step 13.2.11.2.2

Cancel the common factor.

$\frac{x^{2} e^{8 x} + \frac{4 \cdot - 2}{4 \cdot 1} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})}{8} + C$

Step 13.2.11.2.3

Rewrite the expression.

$\frac{x^{2} e^{8 x} + \frac{- 2}{1} (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})}{8} + C$

Step 13.2.11.2.4

Divide $- 2$ by $1$ .

$\frac{x^{2} e^{8 x} - 2 (\frac{x e^{8 x}}{8} - \frac{e^{u}}{64})}{8} + C$

Step 14

Replace all occurrences of $u$ with $8 x$ .

$\frac{x^{2} e^{8 x} - 2 (\frac{x e^{8 x}}{8} - \frac{e^{8 x}}{64})}{8} + C$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$\frac{x^{2} e^{8 x} - 2 \frac{x e^{8 x}}{8} - 2 (- \frac{e^{8 x}}{64})}{8} + C$

Step 15.2

Cancel the common factor of $2$ .

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Step 15.2.1

Factor $2$ out of $- 2$ .

$\frac{x^{2} e^{8 x} + 2 (- 1) \frac{x e^{8 x}}{8} - 2 (- \frac{e^{8 x}}{64})}{8} + C$

Step 15.2.2

Factor $2$ out of $8$ .

$\frac{x^{2} e^{8 x} + 2 \cdot - 1 \frac{x e^{8 x}}{2 \cdot 4} - 2 (- \frac{e^{8 x}}{64})}{8} + C$

Step 15.2.3

Cancel the common factor.

$\frac{x^{2} e^{8 x} + 2 \cdot - 1 \frac{x e^{8 x}}{2 \cdot 4} - 2 (- \frac{e^{8 x}}{64})}{8} + C$

Step 15.2.4

Rewrite the expression.

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} - 2 (- \frac{e^{8 x}}{64})}{8} + C$

Step 15.3

Cancel the common factor of $2$ .

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Step 15.3.1

Move the leading negative in $- \frac{e^{8 x}}{64}$ into the numerator.

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} - 2 \frac{- e^{8 x}}{64}}{8} + C$

Step 15.3.2

Factor $2$ out of $- 2$ .

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} + 2 (- 1) \frac{- e^{8 x}}{64}}{8} + C$

Step 15.3.3

Factor $2$ out of $64$ .

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} + 2 \cdot - 1 \frac{- e^{8 x}}{2 \cdot 32}}{8} + C$

Step 15.3.4

Cancel the common factor.

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} + 2 \cdot - 1 \frac{- e^{8 x}}{2 \cdot 32}}{8} + C$

Step 15.3.5

Rewrite the expression.

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} - \frac{- e^{8 x}}{32}}{8} + C$

Step 15.4

Simplify each term.

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Step 15.4.1

Move the negative in front of the fraction.

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} - - \frac{e^{8 x}}{32}}{8} + C$

Step 15.4.2

Multiply $- - \frac{e^{8 x}}{32}$ .

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Step 15.4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} + 1 \frac{e^{8 x}}{32}}{8} + C$

Step 15.4.2.2

Multiply $\frac{e^{8 x}}{32}$ by $1$ .

$\frac{x^{2} e^{8 x} - \frac{x e^{8 x}}{4} + \frac{e^{8 x}}{32}}{8} + C$

Step 16

Reorder terms.

$\frac{1}{8} (x^{2} e^{8 x} - \frac{1}{4} x e^{8 x} + \frac{1}{32} e^{8 x}) + C$