\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

I got this question by my college professor while I was bunking the class.

The doubt is from Integration by Partial Fractions topic in division Integrals of Mathematics – Class 12

NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options

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Right choice is (a) log|x| – \frac{1}{2} log(x^2+1) + C

Explanation: We know that \int \frac{dx}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}

By simplifying it we get, \int \frac{dx}{x(x^2+1)}=\frac{(A+B) x^2+Cx+A}{x(x^2+1)}

Now equating the coefficients we get A = 0, B = 0, C=1.

\int \frac{dx}{x(x^2+1)} = \int \frac{dx}{x} + \int \frac{-xdx}{(x^2+1)}

Therefore after integrating we get log|x| – \frac{1}{2} log(x^2+1) + C.