A) \[\log (\sec x+\tan x)+2\tan \frac{x}{2}+c\]
B) \[\log (\sec x+\tan x)-2\tan \frac{x}{2}+c\]
C) \[\log (\sec x+\tan x)+\tan \frac{x}{2}+c\]
D) \[\log (\sec x+\tan x)-\tan \frac{x}{2}+c\]
Correct Answer: D
Solution :
\[\int_{{}}^{{}}{\frac{1}{\cos x(1+\cos x)}}dx=\int_{{}}^{{}}{\frac{dx}{\cos x}-\int_{{}}^{{}}{\frac{dx}{1+\cos x}}}\] \[=\int_{{}}^{{}}{\sec x\ dx-\frac{1}{2}\int_{{}}^{{}}{{{\sec }^{2}}\frac{x}{2}dx}}\] \[=\log (\sec x+\tan x)-\tan \frac{x}{2}+c.\]
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