Question

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

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

Answer 1

The correct option is C

6021

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