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

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

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integral of e^{-sqrt(x)}

\int e^{- x} d x

2 (- e^{- x} x - e^{- x}) + C

Solution steps

\int e^{- x} d x

Apply u-substitution

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

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

Apply Integration By Parts

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

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

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

= 2 (e^{u} u - e^{u})

Substitute back

u = - x

= 2 (e^{- x} (- x) - e^{- x})

Simplify

= 2 (- e^{- x} x - e^{- x})

Add a constant to the solution

= 2 (- e^{- x} x - e^{- x}) + C

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