First slide

Methods of integration

Question

If $\int \frac{d x}{x^{4} - x^{2}} = \frac{1}{x} + \log | f (x) | + C$ then $f (x)$ is given by

Moderate

A

$\frac{x + 1}{x - 1}$
B

$\frac{x - 1}{x + 1}$
C

${(\frac{x - 1}{x + 1})}^{1 / 2}$
D

${(\frac{x - 1}{x + 1})}^{2}$

Solution

$\begin{array}{l} \frac{1}{x^{4} - x^{2}} = \frac{1}{x^{2} (x - 1) (x + 1)} = \frac{1}{2} \frac{1}{x^{2}} [\frac{1}{x - 1} - \frac{1}{x + 1}] \\ = \frac{1}{2} [\frac{1}{x - 1} - \frac{1}{x} - \frac{1}{x^{2}} - [\frac{1}{x^{2}} - (\frac{1}{x} - \frac{1}{x + 1})]] \\ = \frac{1}{2} [\frac{1}{x - 1} - \frac{2}{x^{2}} - \frac{1}{x + 1}] \\ \int \frac{d x}{x^{4} - x^{2}} = \frac{1}{2} [\log | x - 1 | - \log | x + 1 | + \frac{2}{x}] + C \\ = \frac{1}{x} + \log {|\frac{x - 1}{x + 1}|}^{1 / 2} + C . \end{array}$

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