A) \[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+c\]
B) \[x{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+c\]
C) \[2[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}]+c\]
D) \[3[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}]+c\]
Correct Answer: D
Solution :
Put \[x=\sin \theta \Rightarrow dx=\cos \theta \,d\theta ,\] therefore \[\int_{{}}^{{}}{{{\sin }^{-1}}(3x-4{{x}^{3}})}\,dx=\int_{{}}^{{}}{{{\sin }^{-1}}(\sin 3\theta )\cos \theta \,d\theta }\] \[=\int_{{}}^{{}}{3\theta \cos \theta \,d\theta }=3\left\{ \theta \sin \theta -\int_{{}}^{{}}{\sin \theta \,d\theta } \right\}\] \[=3\left\{ \theta \sin \theta +\cos \theta \right\}+c=3\left\{ x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}} \right\}+c.\]
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