0-pi-2-cos-x-1-sin-2x-3-dx-

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

\underset{0}{\int^{\frac{π}{2}}} \frac{\cos x}{{(1 + \sqrt{\sin 2 x})}^{3}} d x = ?

Commented by infinityaction last updated on 19/Jun/22

I = \int_{0}^{π / 2} \frac{\cos (π / 2 - x)}{1 + \sqrt{\sin 2 (π / 2 - x)}} d x

2 I = \int_{0}^{π / 2} \frac{\cos x + \sin x}{{[1 + \sqrt{1 - (1 - \sin 2 x)}]}^{3}} d x

1 - \sin 2 x = {(\sin x - \cos x)}^{2}

l e t

p = \sin x - \cos x \to d p = \cos x + \sin x) d x

2 I = \int_{- 1}^{1} \frac{d p}{{(1 + \sqrt{1 - p^{2}})}^{3}}

l e t p = \sin \emptyset \to d p = \cos \emptyset d \emptyset

I = \frac{1}{2} \int_{- π / 2}^{π / 2} \frac{\cos \emptyset}{{(1 + \cos \emptyset)}^{3}} d \emptyset

I = \int_{0}^{π / 2} \frac{\cos \emptyset}{{(1 + \cos \emptyset)}^{3}} d \emptyset

t = \tan (\frac{\emptyset}{2})

I = \int_{0}^{1} (1 - t^{4}) d t

I = \frac{1}{5}