Integrating Square Root of a Rational Function with Variable Substitution

[vee6]

How to integrate below?

$$\int \sqrt{\frac{r^{2}-x^{2}+x^{2}}{r^{2}-x^{2}}} dx$$

neither if u = r^2 - x^2 nor u = x^2 will get the result.

 

[mfb]

Try u=x/r, and simplify, it should become a standard integral (solvable with formulas, or with a trigonometric substitution)

 

[HallsofIvy]

How to integrate below?

$$\int \sqrt{\frac{r^{2}-x^{2}+x^{2}}{r^{2}-x^{2}}} dx$$

neither if u = r^2 - x^2 nor u = x^2 will get the result.

Is this really what you meant? Since -x^2+ x^2= 0 the is exactly the same as
|r|\int \frac{1}{\sqrt{r^2- x^2}}dx

The substitution x= sin(t) would be a stadard technique for this.