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

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

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

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

2 (- e^{- x^{\frac{1}{2}}} x^{\frac{1}{2}} - e^{- x^{\frac{1}{2}}}) + C

Solution steps

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

Apply u-substitution

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

= \int e^{- u} \cdot 2 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^{\frac{1}{2}}

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

Add a constant to the solution

= 2 (- e^{- x^{\frac{1}{2}}} x^{\frac{1}{2}} - e^{- x^{\frac{1}{2}}}) + C

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