What is the integral of (x+y^2)/((x^2+y^2)^{3/2)} ?

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

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integral of (x+y^2)/((x^2+y^2)^{3/2)}

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

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

Solution steps

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

Expand

\frac{y ^{2} + x}{( y ^{2} + x ^{2} ) ^{\frac{3}{2}}} : \frac{y ^{2}}{( y ^{2} + x ^{2} ) ^{\frac{3}{2}}} + \frac{x}{( y ^{2} + x ^{2} ) ^{\frac{3}{2}}}

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

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

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

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

Add a constant to the solution

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

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