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

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

Text Solution

Verified by Experts

`I=int(4x+1)/(x^(2)+3x+2)dx`
`=int(2(2x+3)-5)/(x^(2)+3x+2)dx`
`=2int(2x+3)/(x^(2)+3x+2)dx-5int(1)/(x^(2)+3x+2)dx`
`=2log|x^(2)+3x+2|-5int(1)/(x^(2)+3x+(9//4)-(9//4)+2)dx`
`=2log|x^(2)+3x+2|-5int(1)/((x+3//2)^(2)-(1//2)^(2))dx`
`=2log|x^(2)+3x+2|-5(1)/(2(1//2))log|(x+(3)/(2)-(1)/(2))/(x+(3)/(2)+(1)/(2))|+C`
`=2log|x^(2)+3x+2|-5log|(x+1)/(x+2)|+C`
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