First slide

Methods of integration

Question

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

Moderate

A

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

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

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

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

Solution

$\begin{matrix} Let I = \int \frac{{mx}^{m + 2 n - 1} - {nx}^{n - 1}}{x^{2 m + 2 n} + 2 x^{m + n} + 1} dx = \int \frac{{mx}^{m + 2 n - 1} - {nx}^{n - 1}}{{(1 + x^{m + n})}^{2}} dx \\ ∴ I = \int \frac{{mx}^{m - 1} - \frac{n}{x^{n + 1}}}{{(x^{m} + \frac{1}{x^{n}})}^{2}} dx \\ Let x^{m} + \frac{1}{x^{n}} = t \Rightarrow {mx}^{m - 1} - \frac{n}{x^{n + 1}} dx = dt \\ ∴ I = \int \frac{1}{t^{2}} dt = - \frac{1}{t} + C \\ = \frac{- 1}{(x^{m} + \frac{1}{x^{n}})} + C = \frac{- x^{n}}{x^{m + n} + 1} + C \end{matrix}$

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