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

The integral of-t^2e^{-t} is e^{-t}t^2+2e^{-t}t+2e^{-t}+C

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

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

e^{- t} t^{2} + 2 e^{- t} t + 2 e^{- t} + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

= - (- e^{- t} t^{2} - (- 2 (- e^{- t} t - e^{- t})))

Simplify

- (- e^{- t} t^{2} - (- 2 (- e^{- t} t - e^{- t}))) : e^{- t} t^{2} + 2 e^{- t} t + 2 e^{- t}

= e^{- t} t^{2} + 2 e^{- t} t + 2 e^{- t}

Add a constant to the solution

= e^{- t} t^{2} + 2 e^{- t} t + 2 e^{- t} + C

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