GATE 2026 ES – Question 12
Evaluate $\int\sin x/\sqrt{1+\cos x}\,dx$,where $c$ is an integration constant.
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Correct answer: (B) $-2\sqrt2\cos(x/2)+c$
Explanation
Put $u=1+\cos x$,so $du=-\sin x\,dx$. The integral is $-\int u^{-1/2}du=-2\sqrt{1+\cos x}+c$. Since $1+\cos x=2\cos^2(x/2)$,this becomes $-2\sqrt2|\cos(x/2)|+c$. The official choice **B** uses a branch with $\cos(x/2)>0$; outside such a branch the absolute value/sign must be respected.