Integrate the rational function 1 x^2 - 9 - Vidyadip

Question

Integrate the rational function $\frac{1}{{{x^2} - 9}}$

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Answer

$\int {\frac{1}{{{x^2} - 9}}dx} $

$=\int {\frac{1}{{{x^2} - {3^2}}}dx}$

$= \frac{1}{{2 \times 3}}\log \left| {\frac{{x - 3}}{{x + 3}}} \right| + c$

$\left[ {\because \int {\frac{1}{{{x^2} - {a^2}}}dx = \frac{1}{{2a}}\log \left| {\frac{{x - a}}{{x + a}}} \right|} } \right]$

$= \frac{1}{6}\log \left| {\frac{{x - 3}}{{x + 3}}} \right| + c$.

Which is the required solution.

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