What is the integral from 0 to 0.5 of x^2e^{-x^2} ?

The integral from 0 to 0.5 of x^2e^{-x^2} is 0.03594…

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 0.5 of x^2e^{-x^2}

\int_{0}^{0.5} x^{2} e^{- x^{2}} d x

0.03594 \dots

Solution steps

\int_{0}^{0.5} x^{2} e^{- x^{2}} d x

Simplify

x^{2} : xx

= \int_{0}^{0.5} xx e^{- x^{2}} d x

Apply Integration By Parts

: [\frac{1}{2} (- e^{- x^{2}} x - 2 \cdot \int - \frac{1}{2} e^{- x^{2}} d x)]_{0}^{0.5}

= [\frac{1}{2} (- e^{- x^{2}} x - 2 \cdot \int - \frac{1}{2} e^{- x^{2}} d x)]_{0}^{0.5}

\int - \frac{1}{2} e^{- x^{2}} d x = - \frac{π}{4} erf (x)

= [\frac{1}{2} (- e^{- x^{2}} x - 2 (- \frac{π}{4} erf (x)))]_{0}^{0.5}

Simplify

[\frac{1}{2} (- e^{- x^{2}} x - 2 (- \frac{π}{4} erf (x)))]_{0}^{0.5} : [0.5 (- e^{- x^{2}} x + 0.88622 \dots erf (x))]_{0}^{0.5}

= [0.5 (- e^{- x^{2}} x + 0.88622 \dots erf (x))]_{0}^{0.5}

Compute the boundaries:

0.03594 \dots

= 0.03594 \dots

\int_{- \frac{π}{2}}^{\frac{π}{2}} 2 cos (x) d x

\int_{0}^{t} sin (wt) d t

\int_{1}^{e^{x}} 9 ln (t) d t

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

\int_{- π}^{π} cos^{2} (2 x) d x

What is the integral from 0 to 0.5 of x^2e^{-x^2} ?
The integral from 0 to 0.5 of x^2e^{-x^2} is 0.03594…