`int (dx)/(x(x^(2)+1))` बराबर है:

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

Answer

A

Explanation

माना `(1)/(x(x^(2)+1))=(A)/(x)+(Bx+C)/(x^(2)+1)`
`1=A(x^(2)+1)+(Bx+C)x`
x = 0 तो `1=A(0+1)+0" "rArr" "A=1`
`x^(2)` के गुणांक समान रखने पर,
`0=A+B" "rArr" "B=A=-1`
`x` के गुणांक समान रखने पर , 0 = C
`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`