Question

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

Answer 1

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.