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JEE Main & Advanced Mathematics Definite Integration Question Bank Fundamental definite integration, Definite integration by substitution

  • question_answer
    The correct evaluation of \[\int_{0}^{\pi /2}{\sin x\,\sin 2x}\] is [MP PET 1993, 2003]

    A)                 \[\frac{4}{3}\]   

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

    C)                 \[\frac{3}{4}\]   

    D)                 \[\frac{2}{3}\]

    Correct Answer: D

    Solution :

               Let \[I=\int_{0}^{\pi /2}{\sin x\sin 2x\,dx}=2\int_{0}^{\pi /2}{{{\sin }^{2}}x\cos xdx}\]            Put \[t=\sin x\Rightarrow dt=\cos x\,dx\]                 Now, \[I=2\int_{0}^{1}{{{t}^{2}}dt=\frac{2}{3}[{{t}^{3}}]_{0}^{1}=\frac{2}{3}}\].


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