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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Integration by Substitution

  • question_answer
    \[\int_{{}}^{{}}{{{\sin }^{2}}x\cos x\ dx}\] is equal to [SCRA 1996]

    A)            \[\frac{{{\cos }^{2}}x}{2}+c\]

    B)            \[\frac{{{\sin }^{2}}x}{3}+c\]

    C)            \[\frac{{{\sin }^{3}}x}{3}+c\]

    D)            \[-\frac{{{\cos }^{2}}x}{2}+c\]

    Correct Answer: C

    Solution :

               \[I=\int_{{}}^{{}}{{{\sin }^{2}}x\,.\,\cos x\,dx}\]. Put \[\sin x=t\Rightarrow \cos x\,dx=dt\]            \[\therefore \,\,\,I=\int_{{}}^{{}}{{{t}^{2}}dt}=\frac{{{t}^{3}}}{3}+c=\frac{{{\sin }^{3}}x}{3}+c\].


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