What is the integral from-infinity to infinity of x^2e^{(-x^2)/2} ?

The integral from-infinity to infinity of x^2e^{(-x^2)/2} is sqrt(2)sqrt(pi)

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integral from-infinity to infinity of x^2e^{(-x^2)/2}

\int_{- \infty}^{\infty} x^{2} e^{\frac{- x ^{2}}{2}} d x

2 π

Decimal

2.50662 \dots

Solution steps

\int_{- \infty}^{\infty} x^{2} e^{\frac{- x ^{2}}{2}} d x

Compute the indefinite integral:

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

Compute the boundaries:

\int_{- \infty}^{\infty} x^{2} e^{\frac{- x ^{2}}{2}} d x = \frac{π}{2} - (- \frac{π}{2})

= \frac{π}{2} - (- \frac{π}{2})

Simplify

= 2 π

\frac{d y}{d x}, x y + 4 e^{y} = 4 e

\int \frac{5 - e ^{x}}{e ^{6 x}} d x

\int x^{2} (x^{3} + 5)^{10} d x

y^{^{''}} + 2 y^{^{'}} + y = 3 - e^{- t}

\frac{d}{d x} (sec^{2} (x) + tan^{2} (x))

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