The integration of the function $$\int \sqrt{\frac{r^{2}-x^{2}+x^{2}}{r^{2}-x^{2}}} dx$$ can be simplified using the substitution \( u = \frac{x}{r} \). This leads to a standard integral that can be solved using trigonometric substitution, specifically \( x = r \sin(t) \). The discussion confirms that other substitutions, such as \( u = r^2 - x^2 \) or \( u = x^2 \), do not yield the correct results.
PREREQUISITESStudents and professionals in mathematics, particularly those focusing on calculus and integration techniques, will benefit from this discussion.
Is this really what you meant? Since -x^2+ x^2= 0 the is exactly the same asvee6 said: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.