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