What is the integral of (x^2y+y+2xy) ?

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

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integral of (x^2y+y+2xy)

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

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

Solution steps

\int x^{2} y + y + 2 x y 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

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

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

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

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

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

Add a constant to the solution

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

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