सिद्ध कीजिए कि -
`int(1)/(x[6(logx)^(2)+7 logx+2])dx=log|(1+logx^(2))/(2+logx^(3))|+c`

Explanation

माना `logx=t`
`rArr" "(1)/(x)dx=dt`
माना `" "I=int(1)/(x[6(logx)^(2)+7logx+2])dx`
`" "=int(1)/(6t^(2)+7t+2)dx`
`" "=int(1)/(6t^(2)+7t+2)dx`
`" "=int(1)/((3t+2)(2t+1))dt`
पुनः माना `(1)/((3t+2)(2t+1))=(A)/(3t+2)+(B)/(2t+1)=(A(2t+1)+B(3t+2))/((3t+2)(2t+1))`
`rArr" "A(2t+1)+B(3t+2)=1`
समान घातों के गुणांकों को बराबर रखने पर
`2A+3B=0`
`A+2B=1`
हल करने पर `B=2, A=-3`
`therefore(1)/((3t+2)(2t+1))=(-3)/(3t+2)+(2)/(2t+1)`
और `I=int(-3)/(3t+2)dt+int(2)/(2t+1)dt`
`=-log|3t+2|+log|2t+1|+c`
`=log|(2t+1)/(3t+2)|+c=log|(1+2logx)/(2+3logx)|+c`
`=log|(1+logx^(2))/(2+logx^(3))|+c`