What is the integral of xy(x+y) ?

The integral of xy(x+y) is x((xy^2)/2+(y^3)/3)+C

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integral of xy(x+y)

\int x y (x + y) d y

x (\frac{x y ^{2}}{2} + \frac{y ^{3}}{3}) + C

Solution steps

\int x y (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 (x + y) y d y

Expand

(x + y) y : x y + y^{2}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= x (\int x y d y + \int y^{2} d y)

\int x y d y = \frac{x y ^{2}}{2}

\int y^{2} d y = \frac{y ^{3}}{3}

= x (\frac{x y ^{2}}{2} + \frac{y ^{3}}{3})

Add a constant to the solution

= x (\frac{x y ^{2}}{2} + \frac{y ^{3}}{3}) + C

What is the integral of xy(x+y) ?
The integral of xy(x+y) is x((xy^2)/2+(y^3)/3)+C