A) \[{{A}^{2}}=A\]
B) \[{{(1+A)}^{n}}=I+\lambda A,\]
C) \[\lambda \]
D) None of the above
Correct Answer: D
Solution :
Let \[I=\int_{{}}^{{}}{\frac{1}{\sin \left( x-\frac{\pi }{3} \right)\cos x}}dx\] \[=\frac{1}{\cos \frac{\pi }{3}}\int_{{}}^{{}}{\frac{\cos \left\{ x-\left( x-\frac{\pi }{3} \right) \right\}}{\sin \left( x-\frac{\pi }{3} \right)\cos x}}dx\] \[=2\int_{{}}^{{}}{\frac{\cos x\cos \left( x-\frac{\pi }{3} \right)+\sin x\sin \left( x-\frac{\pi }{3} \right)}{\sin \left( x-\frac{\pi }{3} \right)\cos x}}dx\]\[=2\int_{{}}^{{}}{\left\{ \cot \left( x-\frac{\pi }{3} \right)+\tan x \right\}}dx\] \[=2\left\{ \log \left| \sin \left( x-\frac{\pi }{3} \right) \right|+\log |\sec x| \right\}+C\] \[=2\log \left| \sin \left( x-\frac{\pi }{3} \right)\sec x \right|+C\]
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