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

  • question_answer
    \[\int_{{}}^{{}}{\frac{{{(1+\log x)}^{2}}}{x}}\ dx=\]         [Roorkee 1977]

    A)            \[{{(1+\log x)}^{3}}+c\]

    B)            \[3{{(1+\log x)}^{3}}+c\]

    C)            \[\frac{1}{3}{{(1+\log x)}^{3}}+c\]

    D)            None of these

    Correct Answer: C

    Solution :

                       Put \[(1+\log x)=t\Rightarrow \frac{1}{x}dx=dt\]            \[\int_{{}}^{{}}{\frac{{{(1+\log x)}^{2}}}{x}\,dx=\int_{{}}^{{}}{{{t}^{2}}dt}}\]\[=\frac{{{t}^{2}}}{3}+c=\frac{{{(1+\log x)}^{3}}}{3}+c.\]


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