What is the integral of x^2*e^{xy} ?

The integral of x^2*e^{xy} is xe^{xy}+C

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integral of x^2*e^{xy}

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

x e^{x y} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = x y

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

Simplify

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

= x e^{x y}

Add a constant to the solution

= x e^{x y} + C