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

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

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

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

- \frac{1}{2} e^{- 2 t} - \frac{2}{e ^{t}} + C

Solution steps

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

Expand

\frac{e ^{t} + 2 e ^{2 t}}{e ^{3 t}} : \frac{1}{e ^{2 t}} + \frac{2}{e ^{t}}

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

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

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

= - \frac{1}{2} e^{- 2 t} - \frac{2}{e ^{t}}

Add a constant to the solution

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

\frac{\partial}{\partial x} (\frac{u}{x})

\int_{- 1}^{1} 2 1 - \frac{x ^{2}}{2} d x

\frac{12 x}{x ^{6}}

x \to 1 lim (ln (x^{2}))

f (x) = sin^{4} (3 x + 5)

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