I have following the following eqation,
$$\int_{-1}^{1} \sqrt{(1-a+0.5(1+2b))\left(\frac{1}{r^{6}-1+i 0^{+}}- (a+2b)\right)}dr= \Pi$$
The integral is equal to Pi and this condition determines the possible values of $b$ for values of $a$.
I want to draw a plot of how $b$ vary when $a$ varies from $0.5$ to $1$.
I can't evaluate this integral symbolically due to the singularity. To remove the singularity we can add a small imaginary part to the integration.
$$\int_{-1}^{1} \sqrt{(1-a+0.5(1+2b))\left(\frac{1}{r^{6}-1+i 0^{+}}- (a+2b)\right)}dr= \Pi$$
But even after that, I have the difficulty of finding a way to numerically solve this equation in Mathematica/Matlab to find $a$ and $b$ values.
What are the ways I can approach this problem?