What is the integral of 4x^2e^{-x^2} ?

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

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integral of 4x^2e^{-x^2}

\int 4 x^{2} e^{- x^{2}} d x

- 2 e^{- x^{2}} x + π erf (x) + C

Solution steps

\int 4 x^{2} e^{- x^{2}} d x

Take the constant out:

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

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

Simplify

x^{2} : xx

= 4 \cdot \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

= 4 (- \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)

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

Simplify

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

= - 2 e^{- x^{2}} x + π erf (x)

Add a constant to the solution

= - 2 e^{- x^{2}} x + π erf (x) + C

\frac{3}{x ^{2}}

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