What is the integral of (2x}{e^{\frac{-x^2)/2}} ?

The integral of (2x}{e^{\frac{-x^2)/2}} is 2e^{(x^2)/2}+C

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integral of (2x}{e^{\frac{-x^2)/2}}

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{e ^{\frac{- x ^{2}}{2}}} d x

Simplify

\frac{x}{e ^{\frac{- x ^{2}}{2}}} : e^{\frac{x ^{2}}{2}} x

= 2 \cdot \int e^{\frac{x ^{2}}{2}} x d x

Apply u-substitution

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

= 2 \cdot \int \frac{e ^{\frac{u}{2}}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int e^{\frac{u}{2}} d u

Apply u-substitution

: \int e^{v} \cdot 2 d v

= 2 \cdot \frac{1}{2} \cdot \int e^{v} \cdot 2 d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot 2 \cdot \int e^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= 2 \cdot \frac{1}{2} \cdot 2 e^{v}

Substitute back

= 2 \cdot \frac{1}{2} \cdot 2 e^{\frac{x ^{2}}{2}}

Simplify

2 \cdot \frac{1}{2} \cdot 2 e^{\frac{x ^{2}}{2}} : 2 e^{\frac{x ^{2}}{2}}

= 2 e^{\frac{x ^{2}}{2}}

Add a constant to the solution

= 2 e^{\frac{x ^{2}}{2}} + C

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What is the integral of (2x}{e^{\frac{-x^2)/2}} ?
The integral of (2x}{e^{\frac{-x^2)/2}} is 2e^{(x^2)/2}+C