## 7.2. Integration Techniques

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

We want to again use partial fraction decomposition, so we need to factor the polynomial*p(x) = 1 - x*. Clearly:

^{4}By the partial fraction decomposition theorem we therefore know that:1 - x^{4}= (1 - x^{2}) (1 + x^{2}) = (1 + x) (1 - x) (1 + x^{2})

We can again combine the right side as follows:

so that

for all1 = A (1-x)(1+x^{2}) + B(1+x)(1+x^{2}) + (Cx+D)(1-x^{2}) =

= A + Ax^{2}- Ax - Ax^{3}+ B + Bx^{2}+ Bx + Bx^{3}) + Cx - Cx^{3}+ D - Dx^{2}=

= (A + B + D) + x(-A + B + C) + x^{2}(A + B - D) + x^{3}(-A + B - C)

*x*. This gives four equations in four unknowns:

which - perhaps after reviewing how to efficiently solve systems of linear equations - gives the answer:

A + B + D= 1(for the constant coefficient) -A + B + C = 0(for the xcoefficient)A + B - D = 0(for the xcoefficient)^{2}-A + B - C = 0(for the xcoefficient)^{3}

Therefore we can solve our original integralA = 1/4, B = 1/4, D = 1/2, C = 0