What is the integral of t^3e^{-3t} ?

The integral of t^3e^{-3t} is -1/3 e^{-3t}t^3-1/27 (9e^{-3t}t^2-2(-3e^{-3t}t-e^{-3t}))+C

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integral of t^3e^{-3t}

\int t^{3} e^{- 3 t} d t

- \frac{1}{3} e^{- 3 t} t^{3} - \frac{1}{27} (9 e^{- 3 t} t^{2} - 2 (- 3 e^{- 3 t} t - e^{- 3 t})) + C

Solution steps

\int t^{3} e^{- 3 t} d t

Apply Integration By Parts

= - \frac{1}{3} e^{- 3 t} t^{3} - \int - e^{- 3 t} t^{2} d t

\int - e^{- 3 t} t^{2} d t = \frac{1}{27} (9 e^{- 3 t} t^{2} - 2 (- 3 e^{- 3 t} t - e^{- 3 t}))

= - \frac{1}{3} e^{- 3 t} t^{3} - \frac{1}{27} (9 e^{- 3 t} t^{2} - 2 (- 3 e^{- 3 t} t - e^{- 3 t}))

Add a constant to the solution

= - \frac{1}{3} e^{- 3 t} t^{3} - \frac{1}{27} (9 e^{- 3 t} t^{2} - 2 (- 3 e^{- 3 t} t - e^{- 3 t})) + C

What is the integral of t^3e^{-3t} ?
The integral of t^3e^{-3t} is -1/3 e^{-3t}t^3-1/27 (9e^{-3t}t^2-2(-3e^{-3t}t-e^{-3t}))+C