A) \[\frac{\text{1}}{\text{2}}\text{logtan}\left( \frac{\text{x}}{\text{2}}\text{+}\frac{\text{ }\!\!\pi\!\!\text{ }}{\text{12}} \right)\text{+C}\]
B) \[\frac{\text{1}}{3}\text{logtan}\left( \frac{\text{x}}{\text{2}}-\frac{\text{ }\!\!\pi\!\!\text{ }}{\text{12}} \right)\text{+C}\]
C) \[\text{logtan}\left( \frac{\text{x}}{\text{2}}+\frac{\text{ }\!\!\pi\!\!\text{ }}{6} \right)\text{+C}\]
D) \[\frac{1}{2}\text{logtan}\left( \frac{\text{x}}{\text{2}}-\frac{\text{ }\!\!\pi\!\!\text{ }}{6} \right)\text{+C}\]
Correct Answer: A
Solution :
\[\text{250Jk}{{\text{g}}^{\text{-1}}}{{\text{K}}^{\text{-1}}}\] \[=\frac{1}{2}\int_{{}}^{{}}{\frac{dx}{\frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x}}\] \[=\frac{1}{2}\int_{{}}^{{}}{\frac{dx}{\cos \frac{1}{3}\cos x+\frac{\pi }{3}\sin x}}\] \[\text{U=}\frac{\text{1}}{\text{2}}\text{C}{{\text{V}}^{\text{2}}}\] \[=\frac{1}{2}C\times {{\left( 200 \right)}^{2}}\] \[\text{=2C }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{J}\] \[\Delta \theta \]
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