\(∫\frac {dx}{x(x^2+1)} \ equals \)
\(∫\frac {dx}{x(x^2+1)} \ equals \)
\(log|x|-\frac 12log(x^2+1)+C\)
\(log|x|+\frac 12log(x^2+1)+C\)
\(-log|x|+\frac 12log(x^2+1)+C\)
\(\frac 12log|x|+log(x^2+1)+C\)
The Correct Option is A
Solution and Explanation
Let \(\frac {1}{x(x^2+1)}\) = \(\frac Ax+\frac {Bx+C}{x^2+1}\)
\(1 = A(x^2+1)+(Bx+C)x\)
Equating the coefficients of x2, x, and constant term, we obtain
A + B = 0
C = 0
A = 1
On solving these equations, we obtain
A = 1, B = −1, and C = 0
∴ \(\frac {1}{x(x^2+1)}\) = \(\frac 1x+\frac {-x}{x^2+1}\)
⇒ \(∫\)\(\frac {1}{x(x^2+1)} dx\) = \(∫\)\(\frac 1x-\frac {x}{x^2+1}dx\)
= \(log|x|-\frac 12log|x^2+1|+C\)
Hence, the correct Answer is (A).
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Concepts Used:
Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
