A) \[\tan x-\cot \,x+c\]
B) \[\tan x+\cot \,x+c\]
C) \[\tan x+\operatorname{cosec}\,x+c\]
D) \[\cot x+\operatorname{cosec}\,+c\]
Correct Answer: A
Solution :
\[\int{{{\sec }^{2}}}\,x\,\operatorname{c}\text{ose}{{\text{c}}^{2}}x\,\,dx\] \[=\int{\frac{1}{{{\cos }^{2}}\,x\,{{\sin }^{2}}\,x}}\,dx\] \[=\int{\frac{({{\sin }^{2}}\,x+{{\cos }^{2}}x)\,dx}{{{\sin }^{2}}\,x\,\,{{\cos }^{2}}x}}\] \[=\int{\left( \frac{1}{{{\cos }^{2}}x}+\frac{1}{{{\sin }^{2}}x} \right)}\,\,dx\] \[=\int{{{\sec }^{2}}\,x\,dx+\int{\text{cose}{{\text{c}}^{2}}x\,dx}}\] \[=\tan x-\cot \,x+c\]
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