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

  • question_answer
    \[\int_{{}}^{{}}{\frac{x\ dx}{({{x}^{2}}-{{a}^{2}})({{x}^{2}}-{{b}^{2}})}=}\]              [Roorkee 1976]

    A)                 \[\frac{1}{{{a}^{2}}-{{b}^{2}}}\log \left( \frac{{{x}^{2}}-{{a}^{2}}}{{{x}^{2}}-{{b}^{2}}} \right)+c\]

    B)                 \[\frac{1}{{{a}^{2}}-{{b}^{2}}}\log \left( \frac{{{x}^{2}}-{{b}^{2}}}{{{x}^{2}}-{{a}^{2}}} \right)+c\]

    C)                 \[\frac{1}{2({{a}^{2}}-{{b}^{2}})}\log \left( \frac{{{x}^{2}}-{{a}^{2}}}{{{x}^{2}}-{{b}^{2}}} \right)+c\]          

    D)                 \[\frac{1}{2({{a}^{2}}-{{b}^{2}})}\log \left( \frac{{{x}^{2}}-{{b}^{2}}}{{{x}^{2}}-{{a}^{2}}} \right)+c\]

    Correct Answer: C

    Solution :

                       \[\int_{{}}^{{}}{\frac{x}{({{x}^{2}}-{{a}^{2}})({{x}^{2}}-{{b}^{2}})}}\,dx\]                                 \[=\frac{1}{{{a}^{2}}-{{b}^{2}}}\left[ \int_{{}}^{{}}{\frac{x}{{{x}^{2}}-{{a}^{2}}}\,dx-\int_{{}}^{{}}{\frac{x\,dx}{{{x}^{2}}-{{b}^{2}}}}} \right]\].


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