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Integration by Substitution

Prove that: ...

Question

Prove that:
$\int \frac{1}{4 x^{2} - 1} d x$

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Solution

$\int \frac{1}{4 x^{2} - 1} d x = \int \frac{1}{(2 x)^{2} - 1} d x$
$\Rightarrow$ we know $\int \frac{1}{x^{2} - a^{2}} = \frac{1}{2 a} I n ∣ ∣ \frac{x - a}{x + a} ∣ ∣$
then
$\Rightarrow \frac{1}{2} i n ∣ ∣ \frac{2 x - 1}{2 x + 1} ∣ ∣ + c$

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