What is expand (xe^{-x}+1)e^x ?

The solution to expand (xe^{-x}+1)e^x is x+e^x

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expand (xe^{-x}+1)e^x

expand

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

x + e^{x}

Solution steps

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = e^{x}, b = x e^{- x}, c = 1

= e^{x} x e^{- x} + e^{x} \cdot 1

= e^{x} e^{- x} x + 1 \cdot e^{x}

Simplify

e^{x} e^{- x} x + 1 \cdot e^{x} : x + e^{x}

= x + e^{x}

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