A) \[\frac{-1}{\sin a}\log |\sin \,x\,\text{cosec(x-a) }\!\!|\!\!\text{ +c}\]
B) \[\frac{-1}{\sin a}\log |\sin \,x\,(x-a)\sin x]+c\]
C) \[\frac{-1}{\sin a}\log |\sin \,x\,(x-a)cosecx]+c\]
D) \[\frac{1}{\sin a}\log |\sin \,(\lambda -a)\sin x]+c\]
Correct Answer: A
Solution :
Let \[I=+\int{\text{cosec}\,\,\text{(x-a) cosec x dx}\,}\] \[=\int{\frac{\sin \,a}{\sin \,a\,\sin (x-a)\,\sin x}\,dx}\] \[=-\frac{1}{\sin \,a}\int{\frac{\sin [(x-a)-x]}{\sin (x-a)\sin \,x}\,dx}\] \[=-\frac{1}{\sin \,a}\int{\left[ \frac{\sin (x-a)\,\cos x-\cos (x-a)\,\sin x}{\sin (x-a)\,\sin x} \right]}dx\]\[=-\frac{1}{\sin \,a}\int{[\cot \,x-\,\cot (x-a)]dx}\] \[=-\frac{1}{\sin a}[\log |sin\,x|-log|sin(x-a)|]+c\] \[=\frac{-1}{\sin \,a}[\log |\sin x\,\text{cosec (x-a) }\!\!|\!\!\text{ }\!\!]\!\!\text{ +c}\]
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