A) \[\cot x-x+c\]
B) \[-\cot x+x+c\]
C) \[\cot x+x+c\]
D) \[-\cot x-x+c\]
Correct Answer: D
Solution :
\[\int_{{}}^{{}}{\frac{\cot x\tan x}{{{\sec }^{2}}x-1}\,dx=\int_{{}}^{{}}{\frac{1}{{{\tan }^{2}}x}\,dx=\int_{{}}^{{}}{{{\cot }^{2}}x\,dx}}}\] \[=\int_{{}}^{{}}{(co\text{se}{{\text{c}}^{2}}x-1)\,dx}=-\cot x-x+c.\]
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