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

The integral of x^2(e^{-x})^2 is -1/2 e^{-2x}x^2+1/4 (-2e^{-2x}x-e^{-2x})+C

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integral of x^2(e^{-x})^2

integral

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

- \frac{1}{2} e^{- 2 x} x^{2} + \frac{1}{4} (- 2 e^{- 2 x} x - e^{- 2 x}) + C

Solution steps

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

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

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

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} x^{2} - \int - e^{- 2 x} x d x

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

\int - e^{- 2 x} x d x = - \frac{1}{4} (- 2 e^{- 2 x} x - e^{- 2 x})

= - \frac{1}{2} e^{- 2 x} x^{2} - (- \frac{1}{4} (- 2 e^{- 2 x} x - e^{- 2 x}))

Simplify

= - \frac{1}{2} e^{- 2 x} x^{2} + \frac{1}{4} (- 2 e^{- 2 x} x - e^{- 2 x})

Add a constant to the solution

= - \frac{1}{2} e^{- 2 x} x^{2} + \frac{1}{4} (- 2 e^{- 2 x} x - e^{- 2 x}) + C

1 - x

2 t^{2}

\frac{1}{cos ^{3} ( x )}

\frac{x}{( 1 - x )}

e^{5} x

What is the integral of x^2(e^{-x})^2 ?
The integral of x^2(e^{-x})^2 is -1/2 e^{-2x}x^2+1/4 (-2e^{-2x}x-e^{-2x})+C