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  • .Choose the correct answer integral dx/ x(x^2+1)

Q23   Choose the correct answer \int \frac{dx}{x ( x ^2+1)} \: \: equals

          A ) \log |x| - \frac{1}{2} \log ( x^2 +1 ) + C \\\\ B ) \log |x|+ \frac{1}{2} \log ( x^2 +1 ) + C \\\\ C )- \log |x| + \frac{1}{2} \log ( x^2 +1 ) + C \\\\ D ) \frac{1}{2}\log |x| +\log ( x^2 +1 ) + C

Answers (1)

best_answer

Given integral  \int \frac{dx}{x ( x ^2+1)}

Partial fraction of above equation,

\frac{1}{x ( x ^2+1)} = \frac{A}{x}+\frac{Bx+c}{x^2+1}

1= A(x^2+1)+(Bx+C)x                                                          

Now, equating the coefficients of x^2,x, and the constant term, we get

A+B = 0, C=0, A=1

We have the values, A = 1\ and\ B=-1,\ and\ C=0

\therefore \frac{1}{x ( x ^2+1)} = \frac{1}{x}+\frac{-x}{x^2+1}

\implies \int \frac{1}{x ( x ^2+1)}dx =\int \left \{ \frac{1}{x}+\frac{-x}{x^2+1}\right \}dx

= \log|x| -\frac{1}{2}\log|x^2+1| +C

Therefore, the correct answer is A.

Posted by

Divya Prakash Singh

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