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`

Concept: Integration as an Inverse Process of Differentiation

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