What is the integral of z/(e^{3z)} ?

The integral of z/(e^{3z)} is -1/3 e^{-3z}z-1/9 e^{-3z}+C

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integral of z/(e^{3z)}

\int \frac{z}{e ^{3 z}} d z

- \frac{1}{3} e^{- 3 z} z - \frac{1}{9} e^{- 3 z} + C

Solution steps

\int \frac{z}{e ^{3 z}} d z

Apply Integration By Parts

= - \frac{1}{3} e^{- 3 z} z - \int - \frac{1}{3} e^{- 3 z} d z

\int - \frac{1}{3} e^{- 3 z} d z = \frac{1}{9} e^{- 3 z}

= - \frac{1}{3} e^{- 3 z} z - \frac{1}{9} e^{- 3 z}

Add a constant to the solution

= - \frac{1}{3} e^{- 3 z} z - \frac{1}{9} e^{- 3 z} + C

What is the integral of z/(e^{3z)} ?
The integral of z/(e^{3z)} is -1/3 e^{-3z}z-1/9 e^{-3z}+C