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Question
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
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Solution
Consider the given function
`I=int(x^2+x+1)/((x^2+1)(x+2))dx`
Let `(x^2+x+1)/((x^2+1)(x+2))=A/(x+2)+(Bx+C)/(x^2+1)`
`= (A(x^2+1)+(Bx+C)(x+2))/((x^2+1)(x+2))`
`=((A+B)x^2+(2B+C)x+(2C+A))/((x^2+1)(x+2))`
Thus equating the coefficients, we have,
A+B=1...(1)
2B+C=1...(2)
2C+A=1...(3)
Solving the above three equations, we have,
`A=3/2,B=2/5 `
`:.(x^2+x+1)/((x^2+1)(x+2))=A/(x+2)+(Bx+C)/(x^2+1)`
`=>(x^2+x+1)/((x^2+1)(x+2))=3/(5(x+2))+(2x+1)/(5(x^2+1)`
`:.I=int(x^2+x+1)/((x^2+1)(x+2))dx`
`=int[3/(5(x+2))+(2x+1)/(5(x^2+1))]dx`
`=3/5intdx/((x+2))dx+1/5int(2x+1)/((x^2+1))dx`
`=3/5log(x+2)+1/5int(2x)/((x^2+1))dx+1/5intdx/((x^2+1))`
`=3/2log(x+2)+1/5log(x^2+1)+1/5tan^(-1)x+C`
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