What is the integral of yxe^{xy} ?

The integral of yxe^{xy} is e^{xy}y-(e^{xy})/x+C

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integral of yxe^{xy}

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

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

Solution steps

\int y 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} y d y

Apply Integration By Parts

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

\int e^{x y} \frac{1}{x} d y = e^{x y} \frac{1}{x ^{2}}

= x (e^{x y} \frac{1}{x} y - e^{x y} \frac{1}{x ^{2}})

Simplify

x (e^{x y} \frac{1}{x} y - e^{x y} \frac{1}{x ^{2}}) : e^{x y} y - \frac{e ^{x y}}{x}

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

Add a constant to the solution

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

What is the integral of yxe^{xy} ?
The integral of yxe^{xy} is e^{xy}y-(e^{xy})/x+C