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

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

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integral of (e^{x/2})/(e^x)

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

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

Solution steps

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

Simplify

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

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

(- 1, 2), (1, - 4)

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