Calculus Examples

$\int 2 x^{3} e^{- x^{2}} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{3} e^{- x^{2}} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $- x^{2}$ .

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

Step 2.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- (2 x)$

Step 2.1.4

Multiply $2$ by $- 1$ .

$- 2 x$

Step 2.2

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

$2 \int {\sqrt{- u}}^{2} e^{u} \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Rewrite ${\sqrt{- u}}^{2}$ as $- u$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- u}$ as ${(- u)}^{\frac{1}{2}}$ .

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

Step 3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.3

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

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

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Cancel the common factor.

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

Step 3.1.4.2

Rewrite the expression.

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

Step 3.1.5

Simplify.

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

Step 3.2

Move the negative in front of the fraction.

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

Step 3.3

Multiply $- 1$ by $- 1$ .

$2 \int 1 u e^{u} \frac{1}{2} d u$

Step 3.4

Multiply $u$ by $1$ .

$2 \int u e^{u} \frac{1}{2} d u$

Step 3.5

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

$2 \int \frac{u}{2} e^{u} d u$

Step 3.6

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

$2 \int \frac{u e^{u}}{2} d u$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

$\frac{2}{2} \int u e^{u} d u$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Cancel the common factor.

$\frac{2}{2} \int u e^{u} d u$

Step 5.2.2

Rewrite the expression.

$1 \int u e^{u} d u$

Step 5.3

Multiply $\int u e^{u} d u$ by $1$ .

$\int u e^{u} d u$

Step 6

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

$u e^{u} - \int e^{u} d u$

Step 7

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

$u e^{u} - (e^{u} + C)$

Step 8

Simplify.

$u e^{u} - e^{u} + C$

Step 9

Replace all occurrences of $u$ with $- x^{2}$ .

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