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

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

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integral of 1/2 x^{-1/2}e^{sqrt(x)}

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

e^{x} (x - x + 1) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Simplify

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

= e^{x} (x - x + 1)

Add a constant to the solution

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

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