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Question
Evaluate: `int(5x-2)/(1+2x+3x^2)dx`
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Solution
`int(5x-2)/(1+2x+3x^2)dx=5int(x-2/5)/(1+2x+3x^2)dx=5/6int(6x-12/5)/(1+2x+3x^2)dx`
`=5/6int(6x+2-12/5-2)/(1+2x+3x^2)dx=5/6int(6x+2-22/5)/(1+2x+3x^2)dx`
`=5/6int(6x+2)/(1+2x+3x^2)dx-5/6xx22/5int 1/3{(x+1/3)^2+2/9}dx`
`=5/6 log|1+2x+3x^2|-11/9int 1/3{(x+1/3)^2+2/9}dx`
`=5/6 log|1+2x+3x^2|-11/9xx3/sqrt2 tan^(-1)3/sqrt2(x+1/3)+C`
`=5/6 log|1+2x+3x^2|-11/(3sqrt2) tan^(-1)((3x+1)/sqrt2)+C`
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