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

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

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integral of x^2ye^{xy}

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

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

Solution steps

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

Apply Integration By Parts

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

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

= y (e^{y x} \frac{1}{y} x^{2} - \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}}))

Simplify

y (e^{y x} \frac{1}{y} x^{2} - \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}})) : e^{y x} x^{2} - e^{y x} \frac{2}{y} x + e^{y x} \frac{2}{y ^{2}}

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

Add a constant to the solution

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

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