## 7.2. Integration Techniques

### Example 7.2.12(a): Integrating Rational Functions

The polynomial*p(x)*can be factored into

By the partial fraction decomposition theorem we know that:p(x) = 1 - x^{2}= (1 + x) (1 - x)

for some constants^{1}/_{p(x)}=^{A}/_{1 + x}+^{B}/_{1 - x}

*A*and

*B*. To find these constants, we combine the two rational functions on the right side:

Therefore we must have:

for all1 = A (1 - x) + B (1 + x) = A + B + x(B - A)

*x*. That gives two linear equations in two unknowns:

so thatA + B = 1

B - A = 0

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

(1 - x^{2})^{-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|))]