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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Integration of Rational Function by Using Partial Fractions

  • question_answer
    \[\int_{{}}^{{}}{\frac{dx}{x({{x}^{7}}+1)}}=\]          [Karnataka CET 2004]

    A)                 \[\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\]

    B)                 \[\frac{1}{7}\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\]

    C)                 \[\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\]

    D)                 \[\frac{1}{7}\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\]

    Correct Answer: B

    Solution :

                       Given, \[\int_{{}}^{{}}{\frac{dx}{x\,({{x}^{7}}+1)}}=\int_{{}}^{{}}{\frac{dx}{{{x}^{8}}\left( 1+\frac{1}{{{x}^{7}}} \right)}}\]                    Put \[1+\frac{1}{{{x}^{7}}}=t\] Þ \[\frac{-7}{{{x}^{8}}}dx=dt\]                    \ \[I=\frac{-1}{7}\int{\frac{dt}{t}=}\frac{-1}{7}\log t+c\]                 Þ \[I=-\frac{1}{7}\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\] Þ \[I=\frac{1}{7}\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\].


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