A) \[\sec x-\tan x-\frac{1}{2}\]
B) \[x\sec x+\tan x+\frac{1}{2}\]
C) \[\sec x+x\tan x-\frac{1}{2}\]
D) \[\sec x+\tan x+\frac{1}{2}\]
Correct Answer: D
Solution :
\[\int_{{}}^{{}}{{{e}^{\sec x}}}(sec\,x\,tan\,xf(x)+(sec\,x\,tan\,x+se{{c}^{2}}x)dx\] \[={{e}^{\sec \,x}}f(x)+C,\] Diff. both sides w.r.t. 'x' \[{{e}^{\sec \,x}}(sec\,x\,tan\,\text{x}f(x)+(sec\,x\,tan\,x+{{\sec }^{2}}x))\] \[={{e}^{\sec \,x}}.sec\,x\,tan\,x\,f(x)+{{e}^{\sec x}}f'(x)\] \[f'(x)=se{{c}^{2}}x+\tan x\sec x\] \[\Rightarrow \]\[f(x)=tan\,x+sec\,x+c\]
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