What is the integral from 0 to x of x^2e^{-x} ?

The integral from 0 to x of x^2e^{-x} is -e^{-x}x^2+2(-e^{-x}x-e^{-x})+2

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integral from 0 to x of x^2e^{-x}

\int_{0}^{x} x^{2} e^{- x} d x

- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x}) + 2

Solution steps

\int_{0}^{x} x^{2} e^{- x} d x

Apply Integration By Parts

: [- e^{- x} x^{2} - \int - 2 e^{- x} x d x]_{0}^{x}

= [- e^{- x} x^{2} - \int - 2 e^{- x} x d x]_{0}^{x}

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

= [- e^{- x} x^{2} - (- 2 (- e^{- x} x - e^{- x}))]_{0}^{x}

Simplify

= [- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x})]_{0}^{x}

Compute the boundaries:

- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x}) + 2

= - e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x}) + 2

\int_{0}^{1} (x e^{- x}) d x

\int_{0}^{\frac{1}{3}} 2 (1 - x) d x

\int_{0}^{2} 5^{2 x - 1} - 2^{3 x} d x

\int_{- π}^{π} t^{2} d t

\int_{0}^{2} e^{- r^{2}} d r

What is the integral from 0 to x of x^2e^{-x} ?
The integral from 0 to x of x^2e^{-x} is -e^{-x}x^2+2(-e^{-x}x-e^{-x})+2