Calculus Examples

$x^{2} e^{2 x}$

Step 1

Write $x^{2} e^{2 x}$ as a function.

$f (x) = x^{2} e^{2 x}$

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.

$F (x) = \int x^{2} e^{2 x} d x$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

Multiply $\int e^{2 x} (x) d x$ by $1$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

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

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

Step 11

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

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Step 11.1

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

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Step 11.1.1

Differentiate $2 x$ .

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

Step 11.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 11.1.3

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

$2 \cdot 1$

Step 11.1.4

Multiply $2$ by $1$ .

$2$

Step 11.2

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

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

Step 12

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 14.2

Multiply $2$ by $2$ .

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

Step 15

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

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

Step 16

Simplify.

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Step 16.1

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

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

Step 16.2

Simplify.

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Step 16.2.1

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

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

Step 16.2.2

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

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

Step 16.2.3

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

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

Step 16.2.4

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

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

Step 16.2.5

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

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

Step 16.2.6

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

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

Step 16.2.7

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

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

Step 16.2.8

Combine the numerators over the common denominator.

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

Step 16.2.9

Multiply $2$ by $- 1$ .

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

Step 17

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

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

Step 18

Simplify.

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Step 18.1

Apply the distributive property.

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

Step 18.2

Cancel the common factor of $2$ .

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Step 18.2.1

Factor $2$ out of $- 2$ .

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

Step 18.2.2

Cancel the common factor.

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

Step 18.2.3

Rewrite the expression.

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

Step 18.3

Cancel the common factor of $2$ .

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Step 18.3.1

Move the leading negative in $- \frac{e^{2 x}}{4}$ into the numerator.

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

Step 18.3.2

Factor $2$ out of $- 2$ .

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

Step 18.3.3

Factor $2$ out of $4$ .

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

Step 18.3.4

Cancel the common factor.

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

Step 18.3.5

Rewrite the expression.

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

Step 18.4

Simplify each term.

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Step 18.4.1

Move the negative in front of the fraction.

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

Step 18.4.2

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

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Step 18.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 18.4.2.2

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

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

Step 18.5

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

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

Step 18.6

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

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

Step 18.7

Combine the numerators over the common denominator.

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

Step 18.8

Simplify the numerator.

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Step 18.8.1

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

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Step 18.8.1.1

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

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

Step 18.8.1.2

Multiply by $1$ .

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

Step 18.8.1.3

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

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

Step 18.8.2

Multiply $2$ by $- 1$ .

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

Step 18.9

Factor $- 1$ out of $- 2 x$ .

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

Step 18.10

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 18.11

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

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

Step 18.12

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

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

Step 18.13

Move the negative in front of the fraction.

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

Step 19

Reorder terms.

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

Step 20

The answer is the antiderivative of the function $f (x) = x^{2} e^{2 x}$ .

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