What is the integral of x^2e^{-x^2} ?

The integral of x^2e^{-x^2} is -1/2 e^{-x^2}x+(sqrt(pi))/4 \erf(x)+C

Popular >

integral of x^2e^{-x^2}

integral

x^{2} e^{- x^{2}}

- \frac{1}{2} e^{- x^{2}} x + \frac{π}{4} erf (x) + C

Solution steps

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

Simplify

x^{2} : xx

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{sin ^{4} ( x )}{cos ^{2} ( x )}

\frac{cos ( x )}{x} + \frac{sin ( x )}{x ^{2}}

\frac{2 x + 4}{x ^{3} - 1}

3 x (1 + x)^{- 4}

\frac{u ^{2}}{( u ^{2} + 1 )}

What is the integral of x^2e^{-x^2} ?
The integral of x^2e^{-x^2} is -1/2 e^{-x^2}x+(sqrt(pi))/4 \erf(x)+C