What is expand e^t(te^{-t}+1) ?

The solution to expand e^t(te^{-t}+1) is t+e^t

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expand e^t(te^{-t}+1)

expand

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

t + e^{t}

Solution steps

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

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

Apply the distributive law:

a (b + c) = ab + a c

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

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

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

Simplify

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

= t + e^{t}

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