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

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

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

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

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

Solution steps

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

\int 2 x d y = 2 x y

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

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

Add a constant to the solution

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

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