A) \[\sec x\tan x+\log |\sec x+\tan x|+c\]
B) \[\frac{1}{2}\sec x\tan x+\frac{1}{2}\log |\sec x+\tan x|+c\]
C) \[{{\sec }^{3}}x\tan x+\log |{{\sec }^{3}}x+\tan x|+c\]
D) None of the above
Correct Answer: B
Solution :
Let\[I=\int{{{\sec }^{2}}x}\,\,dx\] \[\Rightarrow \] \[I=\int{\sec x}\cdot {{\sec }^{2}}x\,\,dx\] \[\Rightarrow \] \[I=\sec x\tan x-\int{\sec x\tan x\cdot \tan x\,\,dx}\] (integrating by parts) \[I=\sec x\tan x-\int{\sec x}{{\tan }^{2}}x\,\,dx\] \[=\sec x\tan x-\int{\sec x({{\sec }^{2}}x-1)dx}\] \[\Rightarrow \] \[I=\sec x\tan x-\int{({{\sec }^{2}}x-\sec x)dx}\] \[\Rightarrow \] \[I=\sec x\tan x-\int{{{\sec }^{2}}x\,\,dx+\int{\sec x\,\,dx}}\] \[\Rightarrow \] \[2I=\sec x\tan x+\int{\sec x\,\,dx}\] \[\Rightarrow \] \[2I=\sec x\tan x+\log |\sec x+\tan x|+c\] \[\Rightarrow \] \[I=\frac{1}{2}\sec x\tan x+\frac{1}{2}\log |\sec x+\tan x|+c\]
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