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Integration by Partial Fractions

Prove that : ...

Question

Prove that : $\int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} ln ∣ ∣ ∣ \frac{a + x}{a - x} ∣ ∣ ∣ + c$ .

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Solution

Let $I = \int \frac{1}{a^{2} - x^{2}} d x$
$I = \int \frac{1}{(a - x) (a + x)} d x$
$∴ I = \int \frac{1}{2 a} (\frac{1}{a - x} + \frac{1}{a + x}) d x$ (using partial fractions)
$∴ I = \frac{1}{2 a} (\int \frac{1}{a - x} d x + \int \frac{1}{a + x} d x)$
$∴ I = \frac{1}{2 a} (- ln | a - x | + ln | a + x |) + c$
$∴ I = \frac{1}{2 a} ln ∣ ∣ ∣ \frac{a + x}{a - x} ∣ ∣ ∣ + c$
Hence, proved.

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