What is the integral of (e^y-ye^{-y}) ?

The integral of (e^y-ye^{-y}) is e^y+e^{-y}y+e^{-y}+C

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integral of (e^y-ye^{-y})

\int (e^{y} - y e^{- y}) d y

e^{y} + e^{- y} y + e^{- y} + C

Solution steps

\int e^{y} - y e^{- y} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{y} d y - \int y e^{- y} d y

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

\int y e^{- y} d y = - e^{- y} y - e^{- y}

= e^{y} - (- e^{- y} y - e^{- y})

Simplify

= e^{y} + e^{- y} y + e^{- y}

Add a constant to the solution

= e^{y} + e^{- y} y + e^{- y} + C

What is the integral of (e^y-ye^{-y}) ?
The integral of (e^y-ye^{-y}) is e^y+e^{-y}y+e^{-y}+C