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

The integral of x^2e^{-2x^2} is -1/4 e^{-2x^2}x+(sqrt(pi))/(8sqrt(2))\erf(sqrt(2)x)+C

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

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

- \frac{1}{4} e^{- 2 x^{2}} x + \frac{π}{8 2} erf (2 x) + C

Solution steps

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

Simplify

x^{2} : xx

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

Apply Integration By Parts

: - \frac{1}{4} e^{- 2 x^{2}} x - \int - \frac{1}{4} e^{- 2 x^{2}} d x

= - \frac{1}{4} e^{- 2 x^{2}} x - \int - \frac{1}{4} e^{- 2 x^{2}} d x

\int - \frac{1}{4} e^{- 2 x^{2}} d x = - \frac{π}{8 2} erf (2 x)

= - \frac{1}{4} e^{- 2 x^{2}} x - (- \frac{π}{8 2} erf (2 x))

Simplify

= - \frac{1}{4} e^{- 2 x^{2}} x + \frac{π}{8 2} erf (2 x)

Add a constant to the solution

= - \frac{1}{4} e^{- 2 x^{2}} x + \frac{π}{8 2} erf (2 x) + C

y = 1 - x^{2}, y = 3 - 3 x

\frac{\partial}{\partial x} (2 x y^{4} - 2)

\int (arcsin (x)) d x

\int \frac{sin ( 2 x )}{1 + 3 sin ^{2} ( x )} d x

y^{^{'}} = sin (x + y)

What is the integral of x^2e^{-2x^2} ?
The integral of x^2e^{-2x^2} is -1/4 e^{-2x^2}x+(sqrt(pi))/(8sqrt(2))\erf(sqrt(2)x)+C