2022-01-28

Evaluation of ${\int }_{0}^{\frac{\pi }{2}}\frac{1}{{\left(a{\mathrm{cos}}^{2}\left(x\right)+b{\mathrm{sin}}^{2}\left(x\right)\right)}^{n}}dx$
$n=1,2,3,\dots$
I thought about using but did not work.

Micah May

Expert

Hint:Use Feynman’s Trick: differentiate the integral with respect to the parameters a and b, and it can be shown that:
$\frac{\partial {I}_{n}}{\partial a}+\frac{\partial {I}_{n}}{\partial b}=-n{I}_{n+1}$
This recursion can be re-written alternatively as:
${I}_{n}=-\frac{1}{n-1}\left(\frac{\partial {I}_{n-1}}{\partial a}+\frac{\partial {I}_{n-1}}{\partial b}\right),\phantom{\rule{1em}{0ex}}n=2,3,\dots$
and notice that ${I}_{1}$ can be evaluated rather easily using $u=\mathrm{tan}\left(x\right)$ to get ${I}_{1}=\frac{\pi }{2\sqrt{ab}}$

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