d x x (x^ 2 +1 ) equals - Vidyadip

Question

$\int \frac{d x}{x\left(x^{2}+1\right)}$ equals

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Answer

Let $\frac{1}{x\left(x^{2}+1\right)}=\frac{A}{x}+\frac{B x+C}{x^{2}+1}$
1 = A(X² + 1) + (Bx + C)x = Ax² + A + Bx² + Cx = (A + B)x²+Cx + A
Equating the coefficients of x², x and constant term, we get,
A + B = 0
C = 0
A = 1
On solving these equations, we get,
A = 1, B = -1 and C = 0
Therefore, $\frac{1}{x\left(x^{2}+1\right)}=\frac{1}{x}+\frac{-x}{x^{2}+1}$
$\int \frac{1}{x\left(x^{2}+1\right)}=\int\left\{\frac{1}{x}+\frac{-x}{x^{2}+1}\right\} d x$
= $\log |x|-\frac{1}{2} \log \left|x^{2}+1\right|+C$

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