How do you integrate $(2x)/(x^2-25)$ using partial fractions?

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Answer 1 · 2016-09-02T16:56:38

This could be integrated by substitution, but the question specifies partial fractions, so see below.

Factor the denominator:

$x^2-25 = (x+5)(x-5)$

solve for $A$ and $B$

$A/(x+5)+B/(x-5) = (2x)/(x^2-25)$

$A(x-5)+B(x+5) = 2x$

$Ax-5A+Bx+5B = 2x+0$

$A+B = 2$
$-5A+5B = 0$

$A = B = 1$

$int (2x)/(x^2-25) dx = int (1/(x+5)+1/(x-5)) dx$

$= ln abs(x+5)+ln abs(x-5)+C$

$= ln abs(x^2-25) +C$

How do you integrate (2x)/(x^2-25)(2x)/(x^2-25) using partial fractions?