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

The integral from 0 to y of x^2e^{-xy} is -ye^{-y^2}+(2(-y^2e^{-y^2}-e^{-y^2})+2)/(y^3)

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

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

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

Solution steps

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

Apply Integration By Parts

: [- e^{- y x} \frac{1}{y} x^{2} - \int - e^{- y x} \frac{2}{y} x d x]_{0}^{y}

= [- e^{- y x} \frac{1}{y} x^{2} - \int - e^{- y x} \frac{2}{y} x d x]_{0}^{y}

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

= [- e^{- y x} \frac{1}{y} x^{2} - (- \frac{2}{y ^{3}} (- y e^{- y x} x - e^{- y x}))]_{0}^{y}

Simplify

= [- e^{- y x} \frac{1}{y} x^{2} + \frac{2}{y ^{3}} (- y e^{- y x} x - e^{- y x})]_{0}^{y}

Compute the boundaries:

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

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

Simplify

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

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What is the integral from 0 to y of x^2e^{-xy} ?
The integral from 0 to y of x^2e^{-xy} is -ye^{-y^2}+(2(-y^2e^{-y^2}-e^{-y^2})+2)/(y^3)