What is the integral of ye^{(-y)/x} ?

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

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integral of ye^{(-y)/x}

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

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

Solution steps

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

Simplify

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

Apply Integration By Parts

: - x e^{- \frac{y}{x}} y - \int - x e^{- \frac{y}{x}} d y

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

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

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

Add a constant to the solution

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

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