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Evaluate: int(4x+1)/(x^2+3x+2)dx...

Evaluate: `int(4x+1)/(x^2+3x+2)dx`

Text Solution

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We have, `l=int(4x+1)/(x^(2)+3x+2)dx`
Write, `4x+1=lambda(d)/(dx){x^(2)+3x+2}+mu`
`4x+1=lambda{2x+3}+mu`
`implies4x+1=(2lambda)x+(3lambda+mu)`
On comparison,
`4=2lambdaimplieslambda=2`
and `3lambda+mu=1impliesmu=-5`
`:.l=2intubrace((2x+3)/(x^(2)+3x+2))dx-5intubrace((1)/(x^(2)+3x+2))dx`
put `x^(2)+3x+2=t`
`x^(2)+3x+2=(x+(3)/(2))^(2)-(1)/(4)`
`:.(2x+3)dx=dt`
`:.l=2int(dt)/(t)-5int(1)/((x+(3)/(2))^(2)-(1)/(4))dx`
`l=2log|t|-5log|(x+1)/(x+2)|+C`
`:. l=2log|x^(2)+3x+2|-5log|(x+1)/(x+2)|+C`
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