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

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

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integral of xe^{-xy}

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

- e^{- x y} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = - x y

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

Simplify

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

= - e^{- x y}

Add a constant to the solution

= - e^{- x y} + C

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