A) \[\cos x\log \tan x+\log \,\,\tan (x/2)+c\]
B) \[-\cos x\log \tan x+\log \,\,\tan (x/2)+c\]
C) \[\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\]
D) \[-\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\]
Correct Answer: A
Solution :
| [b] \[\int{\sin x\log (\tan x)dx}\] |
| \[=-\cos x\log \tan x-\int{(-cosx)\frac{1}{\tan x}.{{\sec }^{2}}xdx}\] |
| \[=-\cos x\log \tan x+\int{\frac{1}{\sin x}dx}\] |
| \[=-\cos x\log (\tan x)+\int{\frac{1+{{\tan }^{2}}\frac{x}{2}}{2\tan \frac{x}{2}}dx}\] |
| Now putting \[\frac{x}{2}=t,\] we get, |
| \[=-\cos x\log \tan x+\int{\frac{1}{t}.dt}\] |
| \[=-\cos x\log \tan x+\log (t)+c\] |
| \[=-\cos x\log \tan x+\log \,\,tan\left( \frac{x}{2} \right)+c\] |
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