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\(\int \frac{dx}{x(x^2+1)}\) 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) \(\frac{1}{2} log|x| + log(x^2+1)\) + CI 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»

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.


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