What is the integral of t^2e^{-t/2} ?

The integral of t^2e^{-t/2} is -2e^{-t/2}t^2+4(-2e^{-t/2}t-4e^{-t/2})+C

integral of t^2e^{-t/2}

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

- 2 e^{- \frac{t}{2}} t^{2} + 4 (- 2 e^{- \frac{t}{2}} t - 4 e^{- \frac{t}{2}}) + C

Solution steps

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

Apply Integration By Parts

= - 2 e^{- \frac{t}{2}} t^{2} - \int - 4 e^{- \frac{t}{2}} t d t

\int - 4 e^{- \frac{t}{2}} t d t = - 4 (- 2 e^{- \frac{t}{2}} t - 4 e^{- \frac{t}{2}})

= - 2 e^{- \frac{t}{2}} t^{2} - (- 4 (- 2 e^{- \frac{t}{2}} t - 4 e^{- \frac{t}{2}}))

Simplify

= - 2 e^{- \frac{t}{2}} t^{2} + 4 (- 2 e^{- \frac{t}{2}} t - 4 e^{- \frac{t}{2}})

Add a constant to the solution

= - 2 e^{- \frac{t}{2}} t^{2} + 4 (- 2 e^{- \frac{t}{2}} t - 4 e^{- \frac{t}{2}}) + C

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What is the integral of t^2e^{-t/2} ?
The integral of t^2e^{-t/2} is -2e^{-t/2}t^2+4(-2e^{-t/2}t-4e^{-t/2})+C