How to solve this integral which has a root over the quadratic polynomial divided by another quadratic polynomial?

Question

Can anyone help me solve this integral? I have the value of $b$ (real and positive, most probably lower than $5$) and using the value of integral, I have to calculate $a$.

$$ \int_{0}^{1}\sqrt{\frac{1+ax^2}{1+bx^2}}\,dx.$$

Answer 1

It would be better to know what $b$ is in the original problem, since in full generality

$$ \int_{0}^{1}\sqrt{\frac{1+ax^2}{1+bx^2}}\,dx =\frac{1}{\sqrt{b}}\int_{0}^{\sqrt{b}}\frac{\sqrt{1+\frac{a}{b}x^2}}{\sqrt{1+x^2}}\,dx=\frac{1}{\sqrt{b}}\int_{0}^{\log(\sqrt{b}+\sqrt{b+1})}\sqrt{1+\frac{a}{b}\sinh^2 x}\,dx$$ is just an incomplete elliptic integral of the second kind.

You are looking for the inverse function of a highly non-elementary function. However, the last function, as a function of $a$, behaves like $\sqrt{\frac{a}{b}}$, so the problem is well suited for the application of a numerical method: Newton's method. If $K$ is the given value for the integral, take $a_0$ as $K^2 b$ and

$$ a_{n+1} = a_n + \frac{K-\int_{0}^{1}\sqrt{\frac{1+a_n x^2}{1+b x^2}}\,dx}{\int_{0}^{1}\frac{x^2\,dx}{2\sqrt{(1+a_n x^2)(1+b x^2)}}}.$$ With few iterations, $a_n$ will be an excellent approximation for the wanted $a$. Obviously, you have to compute two incomplete elliptic integrals at every step: choose any quadrature algorithm you like for such a task, that integrand functions are very well-behaved. The state of art is to use the arithmetic-geometric mean, as mentioned by the Wikipedia page about elliptic functions.