A) \[1/2[\sec x\tan x-\log |\sec x+\tan x|]+C\]
B) \[1/2[\sec x\tan x+\log |\sec x-\tan x|]+C\]
C) \[1/2[\sec x\tan x-\log |\sec x-\tan x|]+C\]
D) \[1/2[\sec x\tan x+\log |\sec x+\tan x|]+C\]
Correct Answer: D
Solution :
Let \[l=\int{{{\sec }^{3}}\,x\,dx}\] \[=\int{\underset{I}{\mathop{\sec }}\,\,x{{\underset{II}{\mathop{\sec }}\,}^{2}}}x\,\,dx\] \[\sec x\tan x-\int{\sec x\,ta{{n}^{2}}x\,dx}\] (using integration by parts) \[=\sec x\tan x-\int{\sec x\,({{\sec }^{2}}x-1)\,dx}\] \[\Rightarrow \]\[l=\sec x\tan x-\int{{{\sec }^{3}}\,x\,dx+\int{sex\,x\,\,dx}}\] \[\Rightarrow \] \[l=\sec x\tan x-l+\log |secx+tanx|+2C\] \[\Rightarrow \] \[2l=\sec x\tan x+\log |sex\,x+\tan x|+2C\] \[\Rightarrow \] \[l=\frac{1}{2}[\sec x\,\tan \,x+\log |\sec x+\tan x|]+C\]
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