`int(dx)/(x(x^(2)+1))` is equal to

Options

A. `log|x|-(1)/(2)log(x^(2)+1)+C`
B. `log|x|+(1)/(2)log(x^(2)+1)+C`
C. `-log|x|+(1)/(2)log(x^(2)+1)+C`
D. `(1)/(2)log|x|+log(x^(2)+1)+C`

Answer

A

Explanation

Let `(1)/(x(x^(2)+1))=(A)/(x)+(Bx+C)/(x^(2)+1)`
`rArr" "1=A(x^(2)+1)+(Bx+C)x=(A+B)x^(2)+Cx+A`
On equating the coefficients of `x^(2),x` and constant term on both sides, we get
`A+B=0, C=0` and `A=1`
On solving these equations, we get
`A=1, B=-1 and C=0`
`therefore" "(1)/(x(x^(2)+1))=(1)/(x)+(-x)/(x^(2)+1)`
`therefore" "int(1)/(x(x^(2)+1))dx=int{(1)/(x)-(x)/(x^(2)+1)dx}`
`=log|x|-(1)/(2)log(x^(2)+1)+C`