What is the integral of sqrt(x)e^{-x} ?

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

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integral of sqrt(x)e^{-x}

\int x e^{- x} d x

- e^{- x} x + \frac{π}{2} erf (x) + C

Solution steps

\int x e^{- x} d x

Apply Integration By Parts

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

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

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

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

Simplify

= - e^{- x} x + \frac{π}{2} erf (x)

Add a constant to the solution

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

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