Question

Let $\int \frac{dx}{x^{2008}+x}=\frac{1}{p}\ln \left(\frac{x^q}{1+x^r}\right)+C$ where $p,q,r\in N$ and need not be distinct, then the value of $(p+q+r)$ equals

Answer 1

Answer: 6021

$I=\int \frac{dx}{x\left(x^{2007}+1\right)}$
$=\int \frac{x^{2007}+1-x^{2007}}{x\left(x^{2007}+1\right)}dx$
$=\int \left(\frac{1}{x}-\frac{x^{2006}}{1+x^{2007}}\right)dx$
$=\ln x-\frac{1}{2007}\ln \left(1+x^{2007}\right)$
$=\frac{\ln x^{2007}-\ln \left(1+x^{2007}\right)}{2007}$
$=\frac{1}{2007}\ln \left(\frac{x^{2007}}{1+x^{2007}}\right)+C$
$p+q+r=6021$