A) \[\frac{\pi -2}{4}\]
B) \[\frac{\pi -2}{8}\]
C) \[\frac{\pi -1}{4}\]
D) \[\frac{\pi -1}{2}\]
Correct Answer: C
Solution :
\[I=\int\limits_{0}^{\pi /2}{\frac{{{\sin }^{3}}x}{\sin x+\cos x}}dx\] \[\Rightarrow \]\[I=\int\limits_{0}^{\pi /4}{\frac{{{\sin }^{3}}x+{{\cos }^{3}}x}{\sin x+\cos x}}dx\] \[=\int\limits_{0}^{\pi /4}{(1-sin\,x\,cos\,x)\,}dx\] \[=\left( x-\frac{{{\sin }^{2}}x}{2} \right)_{0}^{\pi /4}\] \[=\frac{\pi }{4}-\frac{1}{4}\] \[=\frac{\pi -1}{4}\]
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