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

The integral of xsqrt(x^2+y^2)-2y is -2yx+1/3 (y^2+x^2)^{3/2}+C

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integral of xsqrt(x^2+y^2)-2y

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

- 2 y x + \frac{1}{3} (y^{2} + x^{2})^{\frac{3}{2}} + C

Solution steps

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

\int 2 y d x = 2 y x

\int y^{2} + x^{2} x d x = \frac{1}{3} (y^{2} + x^{2})^{\frac{3}{2}}

= - 2 y x + \frac{1}{3} (y^{2} + x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - 2 y x + \frac{1}{3} (y^{2} + x^{2})^{\frac{3}{2}} + C

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What is the integral of xsqrt(x^2+y^2)-2y ?
The integral of xsqrt(x^2+y^2)-2y is -2yx+1/3 (y^2+x^2)^{3/2}+C