What is the integral of xe^{-1/2 x^2} ?

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

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integral of xe^{-1/2 x^2}

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

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

Solution steps

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

Simplify

- \frac{1}{2} x^{2} : - \frac{x ^{2}}{2}

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

Apply u-substitution

: \int - e^{u} d u

= \int - e^{u} d u

Take the constant out:

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

= - \int e^{u} d u

Use the common integral:

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

= - e^{u}

Substitute back

u = - \frac{x ^{2}}{2}

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

Add a constant to the solution

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

x \to - \infty lim (e^{a x})

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

\int \frac{1}{x ( x ^{2} - 8 )} d x

f (x) = x (x + 1)^{\frac{1}{3}}

\frac{d x}{d t} = \frac{1}{x + t + 1}

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