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Integration Using Substitution

Solve: log x...

Question

Solve: $log (\frac{x + \sqrt{x^{2} - a^{2}}}{x - \sqrt{x^{2} - a^{2}}})$

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Solution

$l o g (\frac{x + \sqrt{x^{2} - a^{2}}}{x - \sqrt{x^{2} - a^{2}}}) = l o g (\frac{x + \sqrt{x^{2} - a^{2}} (x + \sqrt{x^{2} - a^{2}})}{x - \sqrt{x^{2} - a^{2}} (x + \sqrt{x^{2} - a^{2}})})$
$= l o g (\frac{x^{2} + x^{2} - a^{2} + 2 x \sqrt{x^{2} - a^{2}}}{x^{2} - x^{2} + a^{2}})$
$= l o g (\frac{2 x^{2} - a^{2} + 2 x \sqrt{x^{2} - a^{2}}}{a^{2}}) = l o g (2 (\frac{x}{a})^{2} - 1 + 2 (\frac{x}{a}) {\sqrt{(\frac{x}{a})}}^{2} - 1)$
$∴ l o g (\frac{x + \sqrt{x^{2} - a^{2}}}{x - \sqrt{x^{2} - a^{2}}}) = l o g (2 (\frac{x}{a})^{2} - 1 + 2 (\frac{x}{a}) \sqrt{(\frac{x}{a})^{2}} - 1)$

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