A) \[x\,\log \,(\sin \,x)\]
B) \[-x\,\log \,(\sin \,x)\]
C) \[x\,\log \,(cos\,x)\]
D) \[-x\,\log \,(cos\,x)\]
Correct Answer: B
Solution :
Let \[I=\int_{0}^{x}{\log \,(\cot \,x+\tan t)\,dt}\] \[=\int_{0}^{x}{\log \left( \frac{\cos x}{\sin x}+\frac{\sin t}{\cos t} \right)dt}\] \[=\int_{0}^{x}{[\log \,\{\cos \,(x-t)\}-\log \,\sin \,x-\log \,\cos t]dt}\]\[=\int_{0}^{x}{\log \,\{\cos (x-x+t)\}\,dt}\] \[-\int_{0}^{x}{\log \,\,\sin x\,dt-\int_{0}^{x}{\log \cos \,tdt}}\]\[=\int_{0}^{x}{\log \,\cos t\,dt-[t\,\log \,\sin \,x]_{0}^{x}-\int_{0}^{x}{\log \,\,\cos \,t\,\,dt}}\]\[=-(x\,\log \,\sin x)\]
You need to login to perform this action.
You will be redirected in
3 sec
