Calculus Examples

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

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

Step 1

Let

$u_{2} = e^{- x^{2}}$ . Then

$d u_{2} = - 2 x e^{- x^{2}} d x$ , so

$- \frac{1}{2} d u_{2} = x e^{- x^{2}} d x$ . Rewrite using

$u_{2}$ and

$d$

$u_{2}$ .

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Step 1.1

Let

$u_{2} = e^{- x^{2}}$ . Find

$\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate

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

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

Step 1.1.2

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

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

$g (x) = - x^{2}$ .

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Step 1.1.2.1

To apply the Chain Rule, set

$u_{1}$ as

$- x^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x^{2}]$

Step 1.1.2.2

Differentiate using the Exponential Rule which states that

$\frac{d}{d u_{1}} [a^{u_{1}}]$ is

$a^{u_{1}} \ln (a)$ where

$a$ =

$e$ .

$e^{u_{1}} \frac{d}{d x} [- x^{2}]$

Step 1.1.2.3

Replace all occurrences of

$u_{1}$ with

$- x^{2}$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x^{2}$ with respect to

$x$ is

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

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

Step 1.1.3.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

$e^{- x^{2}} (- (2 x))$

Step 1.1.3.3

Multiply

$2$ by

$- 1$ .

$e^{- x^{2}} (- 2 x)$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Reorder the factors of

$e^{- x^{2}} (- 2 x)$ .

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

Step 1.1.4.2

Reorder factors in

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

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

Step 1.2

Rewrite the problem using

$u_{2}$ and

$d u_{2}$ .

$\int \frac{1}{- 2} d u_{2}$

Step 2

Move the negative in front of the fraction.

$\int - \frac{1}{2} d u_{2}$

Step 3

Apply the constant rule.

$- \frac{1}{2} u_{2} + C$

Step 4

Replace all occurrences of

$u_{2}$ with

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

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