What is the integral of (5e^{5t}+4)/(e^{4t)} ?

The integral of (5e^{5t}+4)/(e^{4t)} is 5e^t-e^{-4t}+C

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integral of (5e^{5t}+4)/(e^{4t)}

\int \frac{5 e ^{5 t} + 4}{e ^{4 t}} d t

5 e^{t} - e^{- 4 t} + C

Solution steps

\int \frac{5 e ^{5 t} + 4}{e ^{4 t}} d t

Expand

\frac{5 e ^{5 t} + 4}{e ^{4 t}} : 5 e^{t} + \frac{4}{e ^{4 t}}

= \int 5 e^{t} + \frac{4}{e ^{4 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 5 e^{t} d t + \int \frac{4}{e ^{4 t}} d t

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

\int \frac{4}{e ^{4 t}} d t = - e^{- 4 t}

= 5 e^{t} - e^{- 4 t}

Add a constant to the solution

= 5 e^{t} - e^{- 4 t} + C

What is the integral of (5e^{5t}+4)/(e^{4t)} ?
The integral of (5e^{5t}+4)/(e^{4t)} is 5e^t-e^{-4t}+C