Question

Use the substitution $x=2\cos hu$ to show that
$\int \frac{1}{(x^2-4)}dx=\cos h^{-1}\left(\frac{x}{2}\right)+constant$

Answer 1

$\int \frac{1}{x^2-4}dx=\int \frac{1}{(x-2)(x+2)}dx$

$\Rightarrow \frac{1}{4}\int \frac{(x+2)-(x-2)}{(x-2)(x+2)}dx$

$\Rightarrow \frac{1}{4}\int (\frac{1}{x-2}-\frac{1}{x+2})dx$

$\Rightarrow \frac{1}{4}\left(ln\right(x-2)-ln(x+2\left)\right)$

$\Rightarrow \frac{1}{4}ln\left(\frac{x-2}{x+2}\right)+constant$

$\Rightarrow {\cosh }^{-1}\left(\frac{x}{2}\right)=constant$