Question

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

Answer 1

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