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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Self Evaluation Test - Integrals

  • question_answer
    If\[\int{\frac{dx}{x({{x}^{n}}+1)}=A\,\,\log \left| \frac{{{x}^{n}}+1}{{{x}^{n}}} \right|+B,B\in R}\]. Then

    A) \[A=\frac{1}{2}\]

    B) \[A=-1\]

    C) \[A=-\frac{1}{n}\]

    D) \[A=\frac{1}{2n}\]

    Correct Answer: C

    Solution :

    [c] \[I=\int{\frac{dx}{x({{x}^{n}}+1)}=\int{\frac{dx}{{{x}^{n+1}}\left( 1+\frac{1}{{{x}^{n}}} \right)}}}\] Put \[1+\frac{1}{{{x}^{n}}}=t\Rightarrow -\frac{n}{{{x}^{n+1}}}dx=dt\] \[I=-\frac{1}{n}\int{\frac{dt}{t}=-\frac{1}{n}ln\,\,t+C=-\frac{1}{n}ln\left( 1+\frac{1}{{{x}^{n}}} \right)+C}\] \[I=-\frac{1}{n}\,\,ln\left( \frac{{{x}^{n}}+1}{{{x}^{n}}} \right)+C\therefore A=-\frac{1}{n}\]


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