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

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

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integral of x/((x^2+y^2)^{1/2)}

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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