What is the integral of (12+3e^{2t})/(e^{2t)} ?

The integral of (12+3e^{2t})/(e^{2t)} is -6e^{-2t}+3t+C

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integral of (12+3e^{2t})/(e^{2t)}

\int \frac{12 + 3 e ^{2 t}}{e ^{2 t}} d t

- 6 e^{- 2 t} + 3 t + C

Solution steps

\int \frac{12 + 3 e ^{2 t}}{e ^{2 t}} d t

Expand

\frac{12 + 3 e ^{2 t}}{e ^{2 t}} : \frac{12}{e ^{2 t}} + 3

= \int \frac{12}{e ^{2 t}} + 3 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 \frac{12}{e ^{2 t}} d t + \int 3 d t

\int \frac{12}{e ^{2 t}} d t = - 6 e^{- 2 t}

\int 3 d t = 3 t

= - 6 e^{- 2 t} + 3 t

Add a constant to the solution

= - 6 e^{- 2 t} + 3 t + C

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What is the integral of (12+3e^{2t})/(e^{2t)} ?
The integral of (12+3e^{2t})/(e^{2t)} is -6e^{-2t}+3t+C