MathCS.org - Real Analysis: Example 7.2.12(a): Integrating Rational Functions

Example 7.2.12(a): Integrating Rational Functions

Find the integral (1 - x²)^-1 dx where the interval [a, b] does not include 1 and -1.

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The polynomial p(x) can be factored into

p(x) = 1 - x² = (1 + x) (1 - x)

By the partial fraction decomposition theorem we know that:

¹ / _p(x) = ^A / _{1 + x} + ^B / _{1 - x}

for some constants A and B. To find these constants, we combine the two rational functions on the right side:

Therefore we must have:

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

for all x. That gives two linear equations in two unknowns:

A + B = 1
B - A = 0

so that A = B = 1/2. Now we can solve our original integral:

(1 - x²)^-1 dx = 1/2 1/ (x + 1) dx - 1/2 1/ (x - 1) dx =
= 1/2 [ (ln(|b + 1|) - ln(|a + 1|)) - (ln(|b - 1|) - ln(|a - 1|))]

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