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Question: \(\frac{1}{x(x^{2} + 1)} = \frac{A}{x} + \frac{Bx + C}{(x^{2} + 1)}\), then \((A,B,C) =\)...

1x(x2+1)=Ax+Bx+C(x2+1)\frac{1}{x(x^{2} + 1)} = \frac{A}{x} + \frac{Bx + C}{(x^{2} + 1)}, then (A,B,C)=(A,B,C) =

A

(1, – 1, 0)

B

(−1,0,−1)( - 1,0, - 1)

C

(0,1,1)(0,1,1)

D

None of these

Answer

(1, – 1, 0)

Explanation

Solution

A(x2+1)+(Bx+C)x=1A(x^{2} + 1) + (Bx + C)x = 1

For x=0,A=1x = 0,A = 1 and for x=ix = i, −B+Ci=1- B + Ci = 1

⇒B=−1,C=0\Rightarrow B = - 1,C = 0 ⇒\Rightarrow (A,B,C)=(1,−1,0)(A,B,C) = (1, - 1,0).