What is the integral of ye^{2xy}+x ?

The integral of ye^{2xy}+x is (x^2)/2+1/2 e^{2yx}+C

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integral of ye^{2xy}+x

\int y e^{2 x y} + x d x

\frac{x ^{2}}{2} + \frac{1}{2} e^{2 y x} + C

Solution steps

\int y e^{2 x y} + x d x

Apply the Sum Rule:

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

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

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

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

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

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{1}{2} e^{2 y x} + C

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