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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    \[\int_{{}}^{{}}{\frac{a\ dx}{b+c{{e}^{x}}}}=\]        [MP PET 1988; BIT Ranchi 1979]

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

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

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

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

    Correct Answer: A

    Solution :

    • \[\int_{{}}^{{}}{\frac{a\,dx}{b+c\,{{e}^{x}}}=\int_{{}}^{{}}{\frac{a{{e}^{x}}}{b{{e}^{x}}+c\,{{e}^{2x}}}\,dx}}\]                   
    • Now put \[{{e}^{x}}=t,\] then it reduces to                   
    • \[a\int_{{}}^{{}}{\frac{dt}{t(ct+b)}=a\int_{{}}^{{}}{-\frac{1}{b}\left\{ \frac{c}{ct+b}-\frac{1}{t} \right\}dt}}\] {By partial fraction}                
    • \[=\frac{a}{b}\log \left( \frac{{{e}^{x}}}{b+c{{e}^{x}}} \right)+c\].


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