What is the integral of (xye^{xy}+2y-1/(x^2)) ?

The integral of (xye^{xy}+2y-1/(x^2)) is -y/(x^2)+y^2+e^{xy}y-(e^{xy})/x+C

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integral of (xye^{xy}+2y-1/(x^2))

\int (x y e^{x y} + 2 y - \frac{1}{x ^{2}}) d y

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

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 2 y d y = y^{2}

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

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

Add a constant to the solution

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

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What is the integral of (xye^{xy}+2y-1/(x^2)) ?
The integral of (xye^{xy}+2y-1/(x^2)) is -y/(x^2)+y^2+e^{xy}y-(e^{xy})/x+C