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  • I have a doubt, kindly clarify. - Integral Calculus - JEE Main-6

\int \frac{dx}{x\left ( x^{n}+1 \right )}=

  • Option 1)

    \frac{1}{n}\log _{e}\left ( \frac{x^{n}}{x^{n}+1} \right )+c

  • Option 2)

    -\frac{1}{n}\log _{e}\left ( \frac{x^{n}+1}{x^{n}} \right )+c

  • Option 3)

    \log _{e}\left ( \frac{x^{n}}{x^{n}+1} \right )+c

  • Option 4)

    none

 

Answers (1)

best_answer

As we learnt

Special type of indefinite integration -

Binomial Differential by substitution :

\int x^{m}(a+bx^{n})^{p}dx

- wherein

Where m,n,p are rational numbers 

 

 Let I=\int \frac{dx}{x\left ( x^{n}+1 \right )}=\int \frac{dx}{x^{n+1}\left ( 1+\frac{1}{x^{n}} \right )}

If \left ( 1+\frac{1}{x^{n}} \right )=p,then\, -\frac{n}{x^{n+1}}dx=dp

ÞI=-\frac{1}{n}\int \frac{dp}{p}= -\frac{1}{n}\log_{e}p+c=-\frac{1}{n}\log_{e}\left ( \frac{x^{n}+1}{x^{n}} \right )+c

 


Option 1)

\frac{1}{n}\log _{e}\left ( \frac{x^{n}}{x^{n}+1} \right )+c

Option 2)

-\frac{1}{n}\log _{e}\left ( \frac{x^{n}+1}{x^{n}} \right )+c

Option 3)

\log _{e}\left ( \frac{x^{n}}{x^{n}+1} \right )+c

Option 4)

none

Posted by

Aadil

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