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

  • question_answer
    \[\int_{{}}^{{}}{\frac{{{x}^{2}}+x-1}{{{x}^{2}}+x-6}\ dx=}\]     [AISSE 1988]

    A)                 \[x+\log (x+3)+\log (x-2)+c\]     

    B)                 \[x-\log (x+3)+\log (x-2)+c\]

    C)                 \[x-\log (x+3)-\log (x-2)+c\]       

    D)                 None of these

    Correct Answer: B

    Solution :

                       \[\int_{{}}^{{}}{\frac{{{x}^{2}}+x-1}{{{x}^{2}}+x-6}\,dx}=\int_{{}}^{{}}{\left[ 1+\frac{5}{{{x}^{2}}+x-6} \right]}\,dx\]                    \[=\int_{{}}^{{}}{\left[ 1+\frac{5}{(x+3)(x-2)} \right]}\,dx\]\[=\int_{{}}^{{}}{dx}+\int_{{}}^{{}}{\frac{dx}{x-2}}-\int_{{}}^{{}}{\frac{dx}{x+3}}\]                 \[=x+\log (x-2)-\log (x+3)+c\].


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