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

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

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integral of xe^{2xy}

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

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

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 2 x y

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

Simplify

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

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

Add a constant to the solution

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