What is the integral of ye^{-xy} ?

The integral of ye^{-xy} is -e^{-yx}+C

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integral of ye^{-xy}

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

- e^{- y x} + C

Solution steps

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

Apply u-substitution

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

= y \cdot \int - \frac{e ^{u}}{y} d u

Take the constant out:

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

= y (- \frac{1}{y} \cdot \int e^{u} d u)

Use the common integral:

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

= y (- \frac{1}{y} e^{u})

Substitute back

u = - y x

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

Simplify

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

= - e^{- y x}

Add a constant to the solution

= - e^{- y x} + C

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