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

The integral of x/(e^{\frac{x^2){2}}} is -1/(e^{\frac{x^2){2}}}+C

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

= \frac{1}{2} (\frac{u}{e ^{\frac{u}{2}}} - e^{- \frac{u}{2}} (u + 2))

Substitute back

u = x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of x/(e^{\frac{x^2){2}}} ?
The integral of x/(e^{\frac{x^2){2}}} is -1/(e^{\frac{x^2){2}}}+C