Calculus Examples

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

Step 1

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = x^{2}$ and

$d v = e^{2 x}$ .

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

Step 2

Simplify.

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Step 2.1

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 2.2

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 3

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 4

Simplify.

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Step 4.1

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 4.2

Cancel the common factor of

$2$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

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 5

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 6

Simplify.

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Step 6.1

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 6.2

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 6.3

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 7

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 8

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 8.1

Let

$u = 2 x$ . Find

$\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate

$2 x$ .

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

Step 8.1.2

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 8.1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$2 \cdot 1$

Step 8.1.4

Multiply

$2$ by

$1$ .

$2$

Step 8.2

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 9

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 10

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 11

Simplify.

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Step 11.1

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 11.2

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 12

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 13

Simplify.

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Step 13.1

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 13.2

Simplify.

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Step 13.2.1

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 13.2.2

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 13.2.3

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 13.2.4

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 13.2.5

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 13.2.6

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 13.2.7

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 13.2.8

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 13.2.9

Multiply

$2$ by

$- 1$ .

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

Step 14

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 15

Simplify.

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Step 15.1

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 15.2

Cancel the common factor of

$2$ .

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Step 15.2.1

Factor

$2$ out of

$- 2$ .

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

Step 15.2.2

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 15.2.3

Rewrite the expression.

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

Step 15.3

Cancel the common factor of

$2$ .

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Step 15.3.1

Move the leading negative in

$- \frac{e^{2 x}}{4}$ into the numerator.

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

Step 15.3.2

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 15.3.3

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 15.3.4

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 15.3.5

Rewrite the expression.

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

Step 15.4

Simplify each term.

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Step 15.4.1

Move the negative in front of the fraction.

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

Step 15.4.2

Multiply

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

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Step 15.4.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 15.4.2.2

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 15.5

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 15.6

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 15.7

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 15.8

Simplify the numerator.

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Step 15.8.1

Factor

$e^{2 x}$ out of

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

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Step 15.8.1.1

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 15.8.1.2

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 15.8.1.3

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 15.8.2

Multiply

$2$ by

$- 1$ .

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

Step 15.9

Factor

$- 1$ out of

$- 2 x$ .

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

Step 15.10

Rewrite

$1$ as

$- 1 (- 1)$ .

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

Step 15.11

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 15.12

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 15.13

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 16

Reorder terms.

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