What is the integral of y*e^{xy} ?

The integral of y*e^{xy} is e^{yx}+C

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integral of y*e^{xy}

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

e^{y x} + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = y x

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

Simplify

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

= e^{y x}

Add a constant to the solution

= e^{y x} + C