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

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

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

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

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

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Solution

The correct option is D $- \frac{x^{q}}{x^{p + q} + 1} + C$
Let $I = \int \frac{p x^{p + 2 q - 1} - q x^{q - 1}}{x^{2 p + 2 q} + 2 x^{p + q} + 1} d x$
$= \int \frac{p x^{p + 2 q - 1} - q x^{q - 1}}{(x^{p + q} + 1)^{2}} d x$
$= \int \frac{p x^{p - 1} - q x^{- q - 1}}{(x^{p} + x^{- q})^{2}} d x$
Let $x^{p} + x^{- q} = t$
$\Rightarrow (p x^{p - 1} - q x^{- q - 1}) d x = d t$
So , $I = \int \frac{d t}{t^{2}}$
$= - \frac{1}{t} + C$
$= - \frac{1}{x^{p} + x^{- q}} + C$
$= - \frac{x^{q}}{x^{p + q} + 1} + C$

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