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

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$\frac{- x^{m}}{x^{m + n} + 1} + C$

$\frac{x^{n}}{x^{m + n} + 1} + C$

$\frac{- x^{n}}{x^{m + n} + 1} + C$

$\frac{x^{m}}{x^{m + n} + 1} + C$

answer is C.

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Detailed Solution

Let $I = \int \frac{m x^{m - 2 n - 1} - n x^{n - 1}}{x^{2 m + 2 n} + 2 x^{m + n} + 1} d x$ . Then

$\begin{array}{l} I = \int \frac{(m x^{m + n} - n) x^{n - 1}}{{(x^{m + n} + 1)}^{2}} d x \\ \Rightarrow I = \int \frac{(m + n) x^{m + 2 n - 1} - n x^{m + 2 n - 1} - n x^{n - 1}}{{(x^{m + n} + 1)}^{2}} d x \\ \Rightarrow I = \int \frac{(m + n) x^{m - n \cdot 1} \times x^{n} - n x^{n}}{{(x^{m + n} + 1)}^{2}} (x^{m + n} + 1) \\ \Rightarrow I = \int x^{n} \times \frac{(m + n) x^{m + n - 1}}{{(x^{m + n} + 1)}^{2}} d x - \int \frac{n x^{n - 1}}{(x^{m + n} + 1)} d x \\ \Rightarrow I = \frac{- x^{n}}{x^{m + n} + 1} - \int n x^{n - 1} \times \frac{- 1}{(x^{m + n} + 1)} - \int \frac{n x^{n - 1}}{(x^{m + n} + 1)} d x + C \end{array}$