Question

If $\int \frac{x^4-x^3+5x^2+x+3}{(x+1)(x^2-x+1{)}^2}$=$log(x+k)+\frac{m}{\sqrt{n}}{\tan }^{-1}\frac{2x-1}{\sqrt{3}}-\frac{1}{x^2-x+1}+\ln (x^2-x-l)+p\int \frac{1}{(x^2-x+1)}$.Find $k+m+n+l+p$?

Answer 1

Let $I=\int \frac{x^4+x^3+5x^2+x+3}{(x+1){(x^2-x+1)}^2}dx$
$\frac{A}{(x+1)}+\frac{Bx+C}{x^2-x+1}+\frac{Dx+E}{{(x^2-x+1)}^2}=\frac{x^4+x^3+{5x}^2+x+3}{{(x+1)(x^2-x+1)}^2}\therefore x^4+x^3+5x^2+x+3=A{(x^2-x+1)}^2+(Bx+C)(x+1)(x^2-x+1)+(Dx+E)(x+1)$
Put $x=1$ we get $9A=9\Rightarrow A=1$
Similarly on solving we get $b=0,C=1,D=2,E=1$
$\therefore I=\int ([\frac{1}{x+1}+\frac{1}{Cx-\frac{1}{2}}]+{\left(\frac{\sqrt{3}}{2}\right)}^2+\frac{2x+1}{(x^2-x+1)}dx)$
$=\log (x+1)+\frac{2}{\sqrt{3}}{\tan }^{-1}\frac{2x-1}{\sqrt{3}}+\int \frac{2x-1+2}{(x^2-x+1)}dx$