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

Explanation

माना `I=int(2x)/((x^(2)+1)(x^(2)+2))dx`
`=int(1)/((t+1)(t+2))dt" माना "x^(2)=t`
`" "rArr 2x dx = dt`
माना `(1)/((t+1)(t+2))=(A)/(t+1)+(B)/(t+2)`
`=(A(t+2)+B(t+1))/((t+1)(t+2))`
`rArr" "A(t+2)+B(t+1)=1`
`t=-1` रखने पर
`A(-1+2)+0=1`
`rArr" "A=1`
`t=-2` रखने पर
`0+B(-2+1)=1`
`rArr" "B=-1`
`therefore" "I=int(1)/(t+1)dt-int(1)/(t+2)dt`
`=log|t+1|-log|t+2|+c`
`=log|(t+1)/(t+2)|+c=log|(x^(2)+1)/(x^(2)+2)|+c`