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

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

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integral of 2y+y/(sqrt(x^2+y^2))

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

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

Solution steps

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

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

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

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

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

Add a constant to the solution

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

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