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

• A. 6024
• B. 6022
• C. 6021
• D. 6020

Answer 1

Answer: (C)

$I=\int \frac{d x}{x\left(x^{2007}+1\right)}=\int \frac{x^{2007}+1-x^{2007}}{x\left(x^{2007}+1\right)} d x=\int\left(\frac{1}{x}-\frac{x^{2006}}{1+x^{2007}}\right) d x$
$=\ln x -\frac{1}{2007} \ln \left(1+ x ^{2007}\right)=\frac{\ln x ^{2007}-\ln \left(1+ x ^{2007}\right)}{2007}=\frac{1}{2007} \ln \left(\frac{ x ^{2007}}{1+ x ^{2007}}\right)+ C$
$p + q + r =6021$
Alternatively: $I=\int \frac{d x}{x^{2008}\left[1+x^{-2007}\right]}=\int \frac{x^{-2008} d x}{1+x^{-2007}}$.