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JEE Main & Advanced Mathematics Definite Integration Question Bank Properties of Definite Integration

  • question_answer
    If \[\int_{\sin x}^{1}{{{t}^{2}}f(t)\ dt=1-\sin x}\],\[x\in \left( 0,\frac{\pi }{2} \right)\] then \[f\ \left( \frac{1}{\sqrt{3}} \right)\]  equal to                                            [IIT Screening 2005]

    A)                 3             

    B)                 \[\frac{1}{3}\]

    C)                 \[\frac{1}{\sqrt{3}}\]     

    D)                 \[\sqrt{3}\]

    Correct Answer: A

    Solution :

               On differentiating both sides, we get                    \[-{{\sin }^{2}}x\ f(\sin x)\cos x=-\cos x\]                                 Þ \[f(\sin x)=\frac{1}{{{\sin }^{2}}x}\]Þ \[f(x)=\frac{1}{{{x}^{2}}}\]Þ \[f\ \left( \frac{1}{\sqrt{3}} \right)=3\].


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