A) \[\frac{2}{3}{{\sin }^{-1}}({{\cos }^{3/2}}x)+c\]
B) \[\frac{3}{2}{{\sin }^{-1}}({{\cos }^{3/2}}x)+c\]
C) \[\frac{2}{3}{{\cos }^{-1}}({{\cos }^{3/2}}x)+c\]
D) None of the above
Correct Answer: C
Solution :
Put\[I=\sqrt{\frac{\cos x(1-{{\cos }^{2}}x)}{1-{{\cos }^{3}}x}}dx\] \[=\int_{{}}^{{}}{\frac{\sqrt{\cos x}\sin x}{\sqrt{1-{{({{\cos }^{3/2}}x)}^{2}}}}}dx\] Put\[{{\cos }^{3/2}}x=t\Rightarrow \frac{3}{2}\sqrt{\cos x}(-\sin x)dx=dt\] \[\therefore \]\[I=\frac{2}{3}\int_{{}}^{{}}{\frac{-dt}{\sqrt{1-{{t}^{2}}}}=\frac{2}{3}{{\cos }^{-1}}(t)+c}\] \[=\frac{2}{3}{{\cos }^{-1}}({{\cos }^{3/2}}x)+c\]
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