What is the integral of xe^{(-x^2)/a} ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of xe^{(-x^2)/a}

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

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

Solution steps

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

Apply u-substitution

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

= \int \frac{e ^{- \frac{u}{a}}}{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 e^{- \frac{u}{a}} d u

Apply u-substitution

: \int - a e^{v} d v

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

Take the constant out:

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

= \frac{1}{2} (- a \cdot \int e^{v} d v)

Use the common integral:

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

= \frac{1}{2} (- a e^{v})

Substitute back

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

Simplify

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

Add a constant to the solution

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

\int y \cdot ln (y) d y

\int 5 (5 - 4 cos (t))^{\frac{1}{4}} sin (t) d t

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

\int 1 + cos (3 x) d x

\int \frac{1}{( x ^{2} + a )} d x

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