A) \[\tan \,x/2\]
B) \[\log \,|\tan \,(x/2)|\]
C) \[\log |\sin \,x|\]
D) \[\log |cos\,x|\]
Correct Answer: B
Solution :
\[\int{\text{cosec x dx=log }\!\!|\!\!\text{ cosec x - cot x }\!\!|\!\!\text{ +c}}\] \[=\log \left| \frac{1}{\sin x}-\frac{\cos x}{\sin x} \right|+c\] \[=\log \left| \frac{(1-\cos \,x)}{\sin \,x} \right|+c\] \[=\log \left| \frac{2{{\sin }^{2}}\frac{x}{2}}{2.\sin \frac{x}{2}.\cos \frac{x}{2}} \right|+c\] \[=\log \left| \tan \frac{x}{2} \right|+c\] On comparing with \[\int{\text{cosec x dx = f(x)+}}\] constant, we get \[f(x)=\log \left| \tan \frac{x}{2} \right|\]
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