A) \[1\]
B) \[2\]
C) \[3\]
D) \[4\]
Correct Answer: B
Solution :
[b] \[I=\int\limits_{0}^{1}{{{\cos }^{-1}}\left( (x-{{x}^{2}})-\sqrt{(1-{{x}^{2}})(2x-{{x}^{2}})} \right)}dx\] \[=\int\limits_{0}^{1}{{{\cos }^{-1}}}\left( x(1-x)-\left( \sqrt{1-{{x}^{2}}} \right)\left( \sqrt{1-{{(1-x)}^{2}}} \right) \right)dx\] \[=\int\limits_{0}^{1}{\left( {{\cos }^{-1}}(x)+{{\cos }^{-1}}(1-x) \right)}dx\] \[=2\int\limits_{0}^{1}{{{\cos }^{-1}}x\,dx}\] \[=2\int\limits_{0}^{\frac{\pi }{2}}{\theta \sin \theta \,\,d\theta }\] (Putting \[x=\cos \theta \]) \[=2\left[ -\theta \cos \theta +\sin \theta \right]_{0}^{\pi /2}=2\]
You need to login to perform this action.
You will be redirected in
3 sec
