What is the integral from 0 to y^3 of xe^{-x/y} ?

The integral from 0 to y^3 of xe^{-x/y} is -y^4e^{-y^2}-y^2e^{-y^2}+y^2

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integral from 0 to y^3 of xe^{-x/y}

\int_{0}^{y^{3}} x e^{- \frac{x}{y}} d x

- y^{4} e^{- y^{2}} - y^{2} e^{- y^{2}} + y^{2}

Solution steps

\int_{0}^{y^{3}} x e^{- \frac{x}{y}} d x

Apply Integration By Parts

: [- y e^{- \frac{x}{y}} x - \int - y e^{- \frac{x}{y}} d x]_{0}^{y^{3}}

= [- y e^{- \frac{x}{y}} x - \int - y e^{- \frac{x}{y}} d x]_{0}^{y^{3}}

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

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

Compute the boundaries:

- y^{4} e^{- y^{2}} - y^{2} e^{- y^{2}} + y^{2}

= - y^{4} e^{- y^{2}} - y^{2} e^{- y^{2}} + y^{2}

\int \frac{24}{1 - cos ( 3 x )} d x

\int \frac{18}{1 - cos ( 9 x )} d x

\frac{2 x 3 x - 1}{5 x}

\int cot^{5} (2 x) d x

\int \frac{4 x ^{2} - 13 x + 12}{x ( x - 2 ) ^{2}} d x

What is the integral from 0 to y^3 of xe^{-x/y} ?
The integral from 0 to y^3 of xe^{-x/y} is -y^4e^{-y^2}-y^2e^{-y^2}+y^2