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

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

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integral of (xy^2+y-x)

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

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

Solution steps

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

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

\int y d x = y x

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

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

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

Add a constant to the solution

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

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