What is the integral of x^{-1/2}e^{-x} ?

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

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integral of x^{-1/2}e^{-x}

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

π erf (x^{\frac{1}{2}}) + C

Solution steps

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

Simplify

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

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

Apply u-substitution

: \int 2 e^{- u^{2}} d u

= \int 2 e^{- u^{2}} d u

Take the constant out:

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

= 2 \cdot \int e^{- u^{2}} d u

Use the nonelementary integral:

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

= 2 \cdot \frac{π}{2} erf (u)

Substitute back

u = x^{\frac{1}{2}}

= 2 \cdot \frac{π}{2} erf (x^{\frac{1}{2}})

Simplify

2 \cdot \frac{π}{2} erf (x^{\frac{1}{2}}) : π erf (x^{\frac{1}{2}})

= π erf (x^{\frac{1}{2}})

Add a constant to the solution

= π erf (x^{\frac{1}{2}}) + C

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