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 $-\frac{x^q}{x^{p+q}+1}+C$


$I=\int \frac{p{x}^{p+2q-1}-q{x}^{q-1}}{(x^{p+q}+1{)}^2}dx=\int \frac{p{x}^{p-1}-q{x}^{-q-1}}{(x^p+x^{-q}{)}^2}dx$
Put $x^p+x^{-q}=t$
$p{x}^{p-1}-q{x}^{-q-1}dx=dt$
Solving we get I = $-\frac{x^q}{x^{p+q}+1}+C$