`int(x^(4))/((x-1)(x^(2)+1))dx` का मान ज्ञात कीजिए ।

Explanation

`(x^(4))/((x-1)(x^(2)+1))=(x^(4))/(x^(3)-x^(2)+x-1)`
`=x+1+(1)/(x^(3)-x^(2)+x-1)`
`=x+1+(1)/((x-1)(x^(2)+x-1)`
`=x+1+(1)/((x-1)(x^(2)+1))`
माना `(1)/((x-1)(x^(2)+1))=(A)/((x-1))+(Bx+C)/((x^(2)+1))`
`therefore" "1=A(x^(2)+1)+(Bx+C)(x-1)" ...(1)"`
समीकरण (1 ) में x = 1 रखने पर
`1=A(1^(2)+1) rArr A=(1)/(2)`
समीकरण (1 ) में `x^(2)` के गुणकों व अचर पदों की तुलना करने पर
`0=A+B " "rArr" "B=-A=-(1)/(2)`
तथा `" "1=A-C" "rArr" "C=A-1=(1)/(2)-1=-(1)/(2)`
इसी प्रकार `(1)/((x+1)(x^(2)+1))=((1)/(2))/((x-1))+(-(1)/(2)x-(1)/(2))/((x^(2)+1))`
`=(1)/(2(x-1))-(1)/(2)((x+1))/((x^(2)+1))`
`rArr" "(x^(4))/((x-1)(x^(2)+1))=(x+1)+(1)/(2(x-1))-(1)/(2)((x+1))/((x^(2)+1))`
`therefore" "int(x^(4))/((x-1)(x^(2)+1))dx=int(x+1)dx+(1)/(2)int(dx)/((x-1))-(1)/(2)int((x+1))/((x^(2)+1))dx`
`=(x^(2))/(2)+x+(1)/(2)log(x-1)-(1)/(2)int(x)/((x^(2)+1))dx-(1)/(2)int(dx)/((x^(2)+1))`
`=(x^(2))/(2)+x+(1)/(2)log(x-1)-(1)/(4)log(x^(2)+1)-(1)/(2)tan^(-1)x`