`(2)/((1-x)(1+x^(2)))`

Explanation

माना `(2)/((1-x)(1+x^(2)))=(A)/(1-x)+(Bx+C)/(1+x^(2))`
`therefore (2)/((1-x)(1+x^(2)))=(A(1+x^(2))+(Bx+C)(1-x))/((1-x)(1+x^(2)))`
`rArr" "2=A(1+x^(2))+(Bx+C)(1-x)`
`{:(x=1,"तो ",2=A(1+1)+0,rArr.,A=1),(x=0,"तो ",2=A+C,,):}`
`x^(2)` के गुणकों को समान रखने पर, `0=A-B`
`rArr" "B=A=1`
`therefore int(2)/((1-x)(1+x^(2)))dx=int{(1)/(1-x)+(x+1)/(x^(2)+1)}dx`
`=int(1)/(1-x)dx+(1)/(2)int(2x)/(x^(2)+1)dx+int(1)/(x^(2)+1)dx`
`=-log|1-x|+(1)/(2)log(1+x^(2))+tan^(-1)x+C_(1)`