Question

$\int \frac{p{x}^{p+2q-1}-q{x}^{q-1}}{x^{2p+2q}+2x^{p+q}+1}dx$ is equal to

• 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

Answer: (C)

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

Let $I=\int \frac{p{x}^{p+2q-1}-q{x}^{q-1}}{x^{2p+2q}+2x^{p+q}+1}dx$
$=\int \frac{p{x}^{p+2q-1}-q{x}^{q-1}}{(x^{p+q}+1{)}^2}dx$
$=\int \frac{p{x}^{p-1}-q{x}^{-q-1}}{(x^p+x^{-q}{)}^2}dx$
Let $x^p+x^{-q}=t$
$\Rightarrow (p{x}^{p-1}-q{x}^{-q-1})dx=dt$
So ,$I=\int \frac{dt}{t^2}$
$=-\frac{1}{t}+C$
$=-\frac{1}{x^p+x^{-q}}+C$
$=-\frac{x^q}{x^{p+q}+1}+C$