What is the integral of x^2ye^{-yx} ?

The integral of x^2ye^{-yx} is -e^{-yx}x^2+(2(-ye^{-yx}x-e^{-yx}))/(y^2)+C

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integral of x^2ye^{-yx}

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

\int - e^{- y x} \frac{2}{y} x d x = - \frac{2}{y ^{3}} (- y e^{- y x} x - e^{- y x})

= y (- e^{- y x} \frac{1}{y} x^{2} - (- \frac{2}{y ^{3}} (- y e^{- y x} x - e^{- y x})))

Simplify

y (- e^{- y x} \frac{1}{y} x^{2} - (- \frac{2}{y ^{3}} (- y e^{- y x} x - e^{- y x}))) : - e^{- y x} x^{2} + \frac{2 ( - y e ^{- y x} x - e ^{- y x} )}{y ^{2}}

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

Add a constant to the solution

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

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What is the integral of x^2ye^{-yx} ?
The integral of x^2ye^{-yx} is -e^{-yx}x^2+(2(-ye^{-yx}x-e^{-yx}))/(y^2)+C