To find:
$$\int \sqrt{1+\sin(x)} dx$$
What I tried:
I put $\tan(\frac{x}{2}) = t$, using which I got it to:
$$I = 2\int \dfrac{1+t}{(1+t^2)^{\frac{3}{2}}}dt$$
Now I am badly stuck. There seems no way to approach this one. Please give a hint. Also, can we initially to some manipulations on the original integral to make it easy? Thank you.
Best Answer
$$(1+\sin x)=\left(\cos\dfrac x2+\sin\dfrac x2\right)^2$$
$$\implies\sqrt{1+\sin x}=\left|\cos\dfrac x2+\sin\dfrac x2\right|$$