Question

Integrate $\sqrt{1+sin( \frac{x}{2} }$. The answer should be of the form : $tan \frac{x}{2} +4(sin \frac{x}{4} - cos \frac{x}{4} )$

Answer 1

I = ∫√1+sin(x/2)
using sin²x+cos²x=1
sin²x/4 +cos²x/4 =1
sinx/2=2sinx/4cosx/4
using these data i can solve this 
I= ∫√sin²x/4+cos²x/4 + 2sinx/4cosx/4dx
it is in the form of (a+b)²= a²+b²+2ab
so using identity 
I= ∫√(sinx/4+cosx/4)²dx
I= ∫sinx/4dx + ∫cosx/4dx
I = -4cosx/4+4sinx/4
I= 4 (sinx/4-cosx/4)+c answer is not in ur form dude