Integral Calculus, Integration by Partial Fractions

Integration by Partial Fractions

In the previous section, we used trig substitution to integrate 1 over 1-x². It was a bit of a chore. In this case there is an easier way. Write the fraction as a linear combination of simpler fractions, i.e. the partial fractions.

1/( 1-x²) = ½ (1/(1+x) + 1/(1-x) )

Now integration is easy, and you get the same formula we saw before.

½ ( log(1+x) - log(1-x) )

In theory, this approach can be used to integrate any rational expression in x. Consider the quotient of two polynomials p(x) over q(x). If the degree of p is at least the degree of q, use synthetic division to write the function as a polynomial plus a proper fraction, where the degree upstairs is strictly less than the degree downstairs. Thus it is enough to consider fractions p/q where the degree of p is less than the degree of q. Furthermore, by pulling the lead coefficients out, we can assume p and q are monic.

The fundamental theorem of algebra tells us q(x), viewed as a polynomial over the reals, can be factored into linear and quadratic pieces. Some of these pieces may appear more than once. For example, let the third "piece" in the factorization of q be w₃ = x-7. If this appears twice in the factorization of q, let v₃ = w₃², or (x-7)². Thus q is the product over v_i.

A technical theorem from ring theory guarantees a unique representation of p/q as a sum of partial fractions whose denominators are v_i. This is what we did above, when separating 1-x² into half of 1/(1+x) + 1/(1-x). It can always be done, and the aforementioned theorem provides a procedure for finding the partial fractions, which can be implemented by a computer. The hard part is factoring q, and representing those factors in a digital format. In practice, we usually apply the method of partial fractions when q is well behaved, e.g. easily factored into small pieces with integer coefficients. Like the quartic 2x⁴+x³-22x²+3x-84, which factors nicely into a quadratic and two linears. In any case, p/q can be rewritten as a sum of partial fractions, and we only need integrate each partial fraction in turn.

If w_i is linear, as in x-7 above, set u = x-7, and integrate with respect to u. Rewrite the numerator in terms of u; it still has the same degree. The result is a linear combination of reciprocal powers of u. Each can be integrated without difficulty.

If w is quadratic, set u = x+c, where c is carefully chosen to make the linear term go away. Now the denominator looks like u² plus or minus a constant. The numerator is a polynomial of degree less than 2, and it can be separated into two terms, u and a constant. These integrals were described in the previous section.

If v_i consists of a quadratic w_i raised to a higher power, an appropriate trig substitution may yet save the day. Or you can revert to complex variables (described below), where each w_i is linear.

Complex Variables

If q is factored over the complex plane, all terms are linear. In fact, the coefficients on q, and on p, could be complex numbers themselves; the theory of partial fractions remains valid. Integration is easy, because, after substitution, each term looks like a reciprocal power of z. When 1/z becomes log(z), you must remove a ray from the complex plane to create a simply connected domain.

If p and q are real, you can recombine complex terms with their conjugates to produce the real integral. Let's illustrate with 1 over x²+1, which has an established integral of atan(x).

Write the integrand as 1/(x+i) - 1/(x-i), all over -2i. The integral becomes log(x+i) - log(x-i), over -2i. Remove the negative x axis, and let the positive x axis have an angle of 0. If x+i has an angle of θ, then x-i has an angle of -θ. The difference between their logs is 2iθ. Divide by -2i and get -θ. One can always adjust an integral by a constant, so call it π/4-θ. This is the angle at the "other" vertex of the triangel, away from the origin, and it is equal to atan(x).