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

The integral of (e^{2t}+e^t)/(e^{2t)+1} is 1/2 ln|e^{2t}+1|+arctan(e^t)+C

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

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

\frac{1}{2} ln e^{2 t} + 1 + arctan (e^{t}) + C

Solution steps

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

Expand

\frac{e ^{2 t} + e ^{t}}{e ^{2 t} + 1} : \frac{e ^{2 t}}{e ^{2 t} + 1} + \frac{e ^{t}}{e ^{2 t} + 1}

Apply the Sum Rule:

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

= \int \frac{e ^{2 t}}{e ^{2 t} + 1} d t + \int \frac{e ^{t}}{e ^{2 t} + 1} d t

\int \frac{e ^{2 t}}{e ^{2 t} + 1} d t = \frac{1}{2} ln e^{2 t} + 1

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

= \frac{1}{2} ln e^{2 t} + 1 + arctan (e^{t})

Add a constant to the solution

= \frac{1}{2} ln e^{2 t} + 1 + arctan (e^{t}) + C

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What is the integral of (e^{2t}+e^t)/(e^{2t)+1} ?
The integral of (e^{2t}+e^t)/(e^{2t)+1} is 1/2 ln|e^{2t}+1|+arctan(e^t)+C