What is the integral from 0 to 1 of xe^{(-x^2)/2} ?

The integral from 0 to 1 of xe^{(-x^2)/2} is 1-1/(sqrt(e))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1 of xe^{(-x^2)/2}

\int_{0}^{1} x e^{\frac{- x ^{2}}{2}} d x

1 - \frac{1}{e}

Decimal Notation

0.39346 \dots

Solution steps

\int_{0}^{1} x e^{\frac{- x ^{2}}{2}} d x

Simplify

= \int_{0}^{1} e^{- \frac{x ^{2}}{2}} x d x

Apply u-substitution

: \int_{0}^{- \frac{1}{2}} - e^{u} d u

= \int_{0}^{- \frac{1}{2}} - e^{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- \frac{1}{2}}^{0} - e^{u} d u

Take the constant out:

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

= - (- \int_{- \frac{1}{2}}^{0} e^{u} d u)

Use the common integral:

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

= - (- [e^{u}]_{- \frac{1}{2}}^{0})

Simplify

= [e^{u}]_{- \frac{1}{2}}^{0}

Compute the boundaries:

1 - \frac{1}{e}

= 1 - \frac{1}{e}

\int_{0}^{5} x^{4} d x

\int_{0}^{\frac{π}{4}} t^{2} sin (4 t) d t

\int_{- 1}^{1} (4 x^{3} - e^{2 x}) d x

\int_{0}^{1} 30 x^{2} (1 - x)^{2} d x

\int_{0}^{1} (3 x^{2} + x) d x

What is the integral from 0 to 1 of xe^{(-x^2)/2} ?
The integral from 0 to 1 of xe^{(-x^2)/2} is 1-1/(sqrt(e))