$$\displaystyle \lim_{n\rightarrow \infty}\frac{\int^{\frac{\pi}{2}}_{0}(\sin x+\cos x)^{n+1}dx}{\int^{\frac{\pi}{2}}_{0}(\sin x+\cos x)^{n}dx}$$
what i try
Let $$I_{n}=2^{\frac{n}{2}}\int^{\frac{\pi}{2}}_{0}\cos^{n+1}\bigg(x-\frac{\pi}{4}\bigg)dx$$
put $\displaystyle x+\frac{\pi}{4}=t$ and $dx=dt$
$$I_{n}=2^{\frac{n}{2}}\int^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\cos^{n+1}(t)dt$$
How do i find $$\Longrightarrow \sqrt{2}\;\; \frac{I_{n}}{I_{n-1}}$$
Help me please
$$I_n = \int_{0}^{\pi/2}(\sin x+\cos x)^n\,dx = 2^{n/2}\int_{-\pi/4}^{\pi/4}\cos^n(t)\,dt=2^{1+n/2}\int_{0}^{\pi/4}\cos^n(t)\,dt$$ differs very little from $$ 2^{1+n/2}\int_{0}^{\pi/2}\cos^n(t)\,dt = 2^{n/2}\sqrt{\pi}\,\frac{\Gamma\left(\frac{1}{2}+\frac{n}{2}\right)}{\Gamma\left(1+\frac{n}{2}\right)} $$ such that the wanted limit is the same as $$ \lim_{n\to +\infty} \sqrt{2}\,\frac{\Gamma\left(1+\frac{n}{2}\right)\Gamma\left(1+\frac{n}{2}\right)}{\Gamma\left(\frac{3}{2}+\frac{n}{2}\right)\Gamma\left(\frac{1}{2}+\frac{n}{2}\right)}=\sqrt{2}.$$
Hint
$$\dfrac{d(\cos^mx\sin x)}{dx}=\cos^{m+1}x-m\cos^{m-1}x(1-\cos^2x)$$
Integrate both sides with respect to $x,$
$$\cos^mx\sin x\big|_{-\pi/4}^{\pi)4}=(m+2)J_{m+1}-mJ_m$$ where $$J_m=\int_{-\pi/4}^{\pi/4}\cos^mx\ dx$$
$$J_1=?,J_0=?$$
