A) \[log(sec\text{ }x+tan\text{ }x)+c\]
B) \[log{{(sec\text{ }x+tan\text{ }x)}^{1/2}}+\text{ }c\]
C) \[~log\,sec\text{ }x\,(sec\text{ }x+tan\text{ }x)+c\]
D) none of the above
Correct Answer: C
Solution :
Let \[I={{\int_{{}}^{{}}{(1+2\tan x(\tan x+\sec x)\}}}^{1/2}}dx\] \[=\int_{{}}^{{}}{{{\{1+2{{\tan }^{2}}x+2\tan x\sec x\}}^{1/2}}}dx\] \[=\int_{{}}^{{}}{{{\{{{\sec }^{2}}x+{{\tan }^{2}}x+2\tan x\sec x\}}^{1/2}}dx}\] \[=\int_{{}}^{{}}{(\sec x+\tan x)dx}\] \[=\log (\sec x+\tan x)+\log \sec x+c\] \[=\log \sec x(\sec x+\tan x)+c\]
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