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Question Number 171392 by cortano1 last updated on 14/Jun/22

$$\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}\frac{\cos x}{{(1+\sqrt{\sin 2x})}^3}dx=?$$

Commented by infinityaction last updated on 19/Jun/22

$$I={\int }_0^{\pi /2}\frac{\cos (\pi /2-x)}{1+\sqrt{\sin 2(\pi /2-x)}}dx$$$$2I={\int }_0^{\pi /2}\frac{\cos x+\sin x}{{[1+\sqrt{1-(1-\sin 2x)}]}^3}dx$$$$1-\sin 2x={(\sin x-\cos x)}^2$$$$let$$$$p=\sin x-\cos x\to dp=\cos x+\sin x)dx$$$$2I={\int }_{-1}^1\frac{dp}{{(1+\sqrt{1-p^2})}^3}$$$$letp=\sin \mathrm{\varnothing }\to dp=\cos \mathrm{\varnothing }d\mathrm{\varnothing }$$$$I=\frac{1}{2}{\int }_{-\pi /2}^{\pi /2}\frac{\cos \mathrm{\varnothing }}{{(1+\cos \mathrm{\varnothing })}^3}d\mathrm{\varnothing }$$$$I={\int }_0^{\pi /2}\frac{\cos \mathrm{\varnothing }}{{(1+\cos \mathrm{\varnothing })}^3}d\mathrm{\varnothing }$$$$t=\tan \left(\frac{\mathrm{\varnothing }}{2}\right)$$$$I={\int }_0^1(1-t^4)dt$$$$I=\frac{1}{5}$$

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