A) \[\frac{1}{12}(7\pi -5)\]
B) \[\frac{1}{12}(7\pi +5)\]
C) \[\frac{3}{10}(4\pi -3)\]
D) \[\frac{3}{20}(4\pi -3)\]
Correct Answer: D
Solution :
\[\int\limits_{-\pi /2}^{-1}{\frac{dx}{2-1}+\int\limits_{-1}^{0}{\frac{dx}{3-1}}+\int\limits_{0}^{1}{\frac{dx}{4}}+\int\limits_{1}^{\pi /2}{\frac{dx}{5+0}}}\] \[=\,\,\,{{(x)}^{-1}}_{-\pi /2}+\frac{1}{2}{{(x)}^{0}}_{-1}+\frac{1}{4}{{(x)}^{1}}_{0}+\frac{1}{5}{{(x)}_{1}}^{\pi /2}\] \[=\,\,\left( -1+\frac{\pi }{2} \right)+\frac{1}{2}(1)+\frac{1}{4}+\frac{1}{5}\left( \frac{\pi }{2}-1 \right)\] \[=\,\,-1+\frac{\pi }{2}+\frac{1}{2}+\frac{1}{4}+\frac{\pi }{10}-\frac{1}{5}\] \[=\,\,-\frac{1}{2}+\frac{1}{4}-\frac{1}{5}+\frac{\pi }{2}+\frac{\pi }{10}\] \[=\,\,\,\frac{-10+5-4}{20}+\frac{5\pi +\pi }{10}\] \[=\,\,-\frac{9}{20}+\frac{6\pi }{10}=-\frac{9}{20}+\frac{3\pi }{5}=\frac{3}{20}(4\pi -3)\]
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