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

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

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integral of ye^{2xy}

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

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

Solution steps

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

Apply u-substitution

= y \cdot \int \frac{e ^{u}}{2 y} d u

Take the constant out:

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

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

Use the common integral:

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

= y \frac{1}{2 y} e^{u}

Substitute back

u = 2 y x

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

Simplify

y \frac{1}{2 y} e^{2 y x} : \frac{1}{2} e^{2 y x}

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

Add a constant to the solution

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