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

The integral of ((1+e^{2t})^2)/(e^{-t)} is e^t+2/3 e^{3t}+1/5 e^{5t}+C

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integral of ((1+e^{2t})^2)/(e^{-t)}

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

e^{t} + \frac{2}{3} e^{3 t} + \frac{1}{5} e^{5 t} + C

Solution steps

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

Expand

\frac{( 1 + e ^{2 t} ) ^{2}}{e ^{- t}} : e^{t} + 2 e^{3 t} + e^{5 t}

= \int e^{t} + 2 e^{3 t} + e^{5 t} d t

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^{t} d t + \int 2 e^{3 t} d t + \int e^{5 t} d t

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

\int 2 e^{3 t} d t = \frac{2}{3} e^{3 t}

\int e^{5 t} d t = \frac{1}{5} e^{5 t}

= e^{t} + \frac{2}{3} e^{3 t} + \frac{1}{5} e^{5 t}

Add a constant to the solution

= e^{t} + \frac{2}{3} e^{3 t} + \frac{1}{5} e^{5 t} + C

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What is the integral of ((1+e^{2t})^2)/(e^{-t)} ?
The integral of ((1+e^{2t})^2)/(e^{-t)} is e^t+2/3 e^{3t}+1/5 e^{5t}+C