Crazy $\int_0^{\frac\pi{2}}\frac {\cos\left((1-2n)\arcsin\left(\frac{\sin(\theta)}{\sqrt 2}\right)\right)}{\sqrt{1-\frac{\sin^2 \theta}{2}}}d\theta$

Question

$$\int_0^{\frac\pi{2}}\frac {\cos\left((1-2n)\arcsin\left(\frac{\sin(\theta)}{\sqrt 2}\right)\right)}{\sqrt{(1-\frac{\sin^2 \theta}{2})}}d\theta$$

I tried using some sort of substitutions but I think this must have some other way to solve and gave me another different integral and gamma functions and all
which now I'm uncertain if it's my cup of tea!

Answer 1

$$\begin{align*}\int_0^{\frac\pi{2}}\frac {\cos\left((1-2n)\arcsin\left(\frac{\sin(\theta)}{\sqrt 2}\right)\right)}{\sqrt{(1-\frac{\sin^2 \theta}{2})}}d\theta & = \frac{\pi}{2}{_2F_1}{(1-n, n;1;\frac 1{2})}\\ & = \frac {\pi}{2}\frac {\sqrt \pi}{\Gamma(1-\frac n{2})\Gamma(\frac 1{2} + \frac n{2})}\\& = \frac {\pi}{2}P_{-n}(1-2x)|_{x = \frac 1{2}}\\ \end{align*}$$

---

• In reply to the comment: $\phi = \sin^{-1}(\sqrt x \sin(\theta))$ $$\left|\int_0^{\sin^{-1}\sqrt x} \frac {\cos\left((1-2n)\phi\right)}{\sqrt{(x-\sin^2\phi)}}d\phi = \frac {\pi}{2}{_2F_1}(1-n,n;1;x)\right|_{x = \frac 1{2}}$$

---

• #Curiosity $$(1-y^2)^{-1/2}\cos(2n\sin^{-1} y) ={_2F_1}(\frac 1{2}+n, \frac 1{2}-n;\frac 1{2};y^2)$$ $(y, 2n) ≡ (\sqrt x \sin \theta, 1-2n)$ & integrating w.r.t $\theta$ over $(0, \pi/2)$