Question Number 186246 by Rupesh123 last updated on 02/Feb/23
Answered by Frix last updated on 02/Feb/23
![∫_0 ^(π/2) (dx/( (√(sin x (1+cos x)))))=^(t=tan (x/2)) =∫_0 ^1 (dt/( (√t)))=2[(√t)]_0 ^1 =2](Q186251.png)
$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{{dx}}{\:\sqrt{\mathrm{sin}\:{x}\:\left(\mathrm{1}+\mathrm{cos}\:{x}\right)}}\overset{{t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}} {=} \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{dt}}{\:\sqrt{{t}}}=\mathrm{2}\left[\sqrt{{t}}\right]_{\mathrm{0}} ^{\mathrm{1}} =\mathrm{2} \\ $$