# Integrating by partial fractions

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• March 18th 2009, 05:49 PM
madmartigano
Integrating by partial fractions
I need to use partial fractions to derive the integration formula

$\int {\frac{{dx}}{{{a^2} - {x^2}}}} = \frac{1}{{2a}}\ln \left| {\frac{{a + x}}{{a - x}}} \right| + C$

So far I've broken the equation into

$\frac{1}{{{a^2} - {x^2}}} = \frac{A}{{a + x}} + \frac{B}{{a - x}}$

$1 = A(a - x) + B(a + x)$

I'm trying to determine the coefficients of A and B, but I can't figure out how to proceed with the derivation from here.
• March 18th 2009, 05:54 PM
skeeter
Quote:

Originally Posted by madmartigano
I need to use partial fractions to derive the integration formula

$\int {\frac{{dx}}{{{a^2} - {x^2}}}} = \frac{1}{{2a}}\ln \left| {\frac{{a + x}}{{a - x}}} \right| + C$

So far I've broken the equation into

$\frac{1}{{{a^2} - {x^2}}} = \frac{A}{{a + x}} + \frac{B}{{a - x}}$

$1 = A(a - x) + B(a + x)$

I'm trying to determine the coefficients of A and B, but I can't figure out how to proceed with the derivation from here.

using your last equation ...

let $x = a$ , solve for B

then let $x = -a$ , solve for A