Evaluate: `int(5x^2+20 x+6)/(x^3+2x^2+x)\ dx`

Explanation

`I=int frac{5 x^{2}+20 x+6}{x^{3}+2 x^{2}+x} `

`=int frac{(5 x^{2}+20 x+6) d x}{x(x^{2}+2 x+1)} `

`=int frac{(5 x^{2}+20 x+6) d x}{x(x+1)^{2}} `

`text { Let } frac{5 x^{2}+20 x+6}{x(x+1)^{2}}=frac{A}{x}+frac{B}{x+1}+frac{C}{(x+1)^{2}} `

`Rightarrow frac{5 x^{2}+20 x+6}{x(x+1)^{2}}=frac{A(x+1)^{2}+B(x)(x+1)+C(x)}{x(x+1)^{2}} `

`Rightarrow 5 x^{2}+20 x+6=A(x^{2}+2 x+1)+B(x^{2}+x)+C x `

`Rightarrow 5 x^{2}+20 x+6=(A+B) x^{2}+(2 A+B+C) x+A`

Equating coefficients of like terms

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