What is the integral of e^{-y/x} ?

The integral of e^{-y/x} is -xe^{-y/x}+C

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integral of e^{-y/x}

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

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

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

= - x \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= - x e^{u}

Substitute back

u = - \frac{y}{x}

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

Add a constant to the solution

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