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

The integral of x^2*e^{-x} is -e^{-x}x^2+2(-e^{-x}x-e^{-x})+C

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integral of x^2*e^{-x}

integral

x^{2} \cdot e^{- x}

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{1}{( 1 + sin ^{2} ( x ) )}

x - 2 sin (x)

\frac{x - 1}{( x + 1 ) ^{3}}

\frac{cos ^{2} ( t )}{2}

\frac{1}{( e ^{x} + e ^{- x} ) ^{2}}

What is the integral of x^2*e^{-x} ?
The integral of x^2*e^{-x} is -e^{-x}x^2+2(-e^{-x}x-e^{-x})+C