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Evlauate : int (1)/(x[6(logx)^(2)+7logx+...

Evlauate : `int (1)/(x[6(logx)^(2)+7logx+2]] dx= log |(1+log x^(2))/(2+log x^(3))|+c`

Text Solution

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`"Let log " x=t`
`rArr " "1/x dx=dt`
` "Now Let " I = int (1)/(x[6(log x )^(2) +7log x+2]]dx`
`=int (1)/(6t^(2) +7t+2)dt`
`=int(1)/((3t+2)(2t+1))dt`
`" Agtain Let " (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)=t`
comparing the coefficients of equal powers
`2A +3B=0`
`A+ 2B =1`
On solving `B=2,A =-3`

`" and "" "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+2log x)/(2+3log x)| +c`
`=log |(1+log x^(2))/(2+log x^(3))| +c.`
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