Question

The evaluation of $\int \frac{p{x}^{p+2q-1}-q{x}^{q-1}}{x^{2p+2q}+2x^{p+q}+1}dx$ is

• A. $-\frac{x^p}{x^{p+q}+1}+C$
• B. $\frac{x^q}{x^{p+q}+1}+C$
• C. $-\frac{x^q}{x^{p+q}+1}+C$
• D. $\frac{x^p}{x^{p+q}+1}+C$

Answer 1

The correct option is C $-\frac{x^q}{x^{p+q}+1}+C$

$I=\int \frac{{px}^{p+2q-1}-q{x}^{q-1}}{x^{2p+2q}+{2x}^{p+q}+1}dx=\int \frac{{px}^{p-1}-q{x}^{-q-1}}{{(x^p+x^{-q})}^2}dx.$
Substitute $x^q+x^{-q}=t\Rightarrow ({px}^{p-1}-{qx}^{-q-1})dx=dt$
$I=\int \frac{1}{t}=-\frac{1}{x^p+x^{-q}}=-\frac{x^q}{x^{p+q}+1.}$