Partial fractions in integration

From Academic Kids

In integral calculus, the use of partial fractions is required to integrate the general rational function. Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of partial fractions. Each partial fraction has as its denominator a polynomial function of degree 1 or 2, or some positive integer power of such a function. If the denominator is a 1st-degree polynomial or a power of such a polynomial, then the numerator is a constant. If the denominator is a 2nd-degree polynomial or a power of such a polynomial, then the numerator is a 1st-degree polynomial.

For an account of how to find this partial fraction expansion of a rational function, see partial fraction.

This article is about what to do after finding the partial fraction expansion, when one is trying to find the function's antiderivative.

Contents

A 1st-degree polynomial in the denominator

The substitution u = ax + b, du = a dx reduces the integral

<math>\int {1 \over ax+b}\,dx<math>

to

<math>\int {1 \over u}\,{du \over a}={1 \over a}\int{du\over u}={1 \over a}\ln\left|u\right|+C = {1 \over a} \ln\left|ax+b\right|+C.<math>

A repeated 1st-degree polynomial in the denominator

The same substitution reduces such integrals as

<math>\int {1 \over (ax+b)^8}\,dx<math>

to

<math>\int {1 \over u^8}\,{du \over a}={1 \over a}\int u^{-8}\,du = {1 \over a} \cdot{u^{-7} \over(-7)} = {-1 \over 7au^7}+C = {-1 \over 7a(ax+b)^7}+C. <math>

An irreducible 2nd-degree polynomial in the denominator

Next we consider such integrals as

<math>\int {x+6 \over x^2-8x+25}\,dx.<math>

The quickest way to see that the denominator x2 − 8x + 25 is irreducible is to observe that its discriminant is negative. Alternatively, we can complete the square:

<math>x^2-8x+25=(x^2-8x+16)+9=(x-4)^2+9\,<math>

and observe that this sum of two squares can never be 0 while x is a real number.

In order to make use of the substitution

<math>u=x^2-8x+25\,<math>
<math>du=(2x-8)\,dx<math>
<math>du/2=(x-4)\,dx<math>

we would need to find x − 4 in the numerator. So we decompose the numerator x + 6 as (x − 4) + 10, and we write the integral as

<math>\int {x-4 \over x^2-8x+25}\,dx + \int {10 \over x^2-8x+25}\,dx.<math>

The substitution handles the first summand, thus:

<math>\int {x-4 \over x^2-8x+25}\,dx = \int {du/2 \over u}

= {1 \over 2}\ln\left|u\right|+C = {1 \over 2}\ln(x^2-8x+25)+C.<math>

Note that the reason we can discard the absolute value sign is that, as we observed earlier, (x − 4)2 + 9 can never be negative.

Next we must treat the integral

<math>\int {10 \over x^2-8x+25} \, dx.<math>

First, complete the square, then do a bit more algebra:

<math>\int {10 \over x^2-8x+25} \, dx

= \int {10 \over (x-4)^2+9} \, dx = \int {10/9 \over \left({x-4 \over 3}\right)^2+1}\,dx<math>

Now the substitution

<math>w=(x-4)/3\,<math>
<math>dw=dx/3\,<math>

gives us

<math>{10 \over 3}\int {dw \over w^2+1}

= {10 \over 3} \arctan(w)+C={10 \over 3} \arctan\left({x-4 \over 3}\right)+C.<math>

A repeated irreducible 2nd-degree polynomial in the denominator

Next, consider

<math>\int {x+6 \over (x^2-8x+25)^{8}}\,dx.<math>

Just as above, we can split x + 6 into (x − 4) + 10, and treat the part containing x − 4 via the substitution

<math>u=x^2-8x+25,\,<math>
<math>du=(2x-8),\,dx<math>
<math>du/2=(x-4)\,dx.<math>

This leaves us with

<math>\int {10 \over (x^2-8x+25)^{8}}\,dx.<math>

As before, we first complete the square and then do a bit of algebraic massaging, to get

<math>\int {10 \over (x^2-8x+25)^{8}}\,dx

=\int {10 \over ((x-4)^2+9)^{8}}\,dx =\int {10/9^{8} \over \left(\left({x-4 \over 3}\right)^2+1\right)^8}\,dx.<math>

The we can use a trigonometric substitution:

<math>\tan\theta={x-4 \over 3},\,<math>
<math>\left({x-4 \over 3}\right)^2+1=\tan^2\theta+1=\sec^2\theta,\,<math>
<math>d\tan\theta=\sec^2\theta\,d\theta={dx \over 3}.\,<math>

Then the integral becomes

<math>\int {30/9^{8} \over \sec^{16}\theta} \sec^2\theta \,d\theta

={30 \over 9^{8}}\int \cos^{14} \theta \, d\theta<math>

By repeated applications of the half-angle formula

<math>\cos^2\theta={1 \over 2}+{1 \over 2} \cos(2\theta)\,<math>

one can reduce this to an integral involving no higher powers of cos θ higher than the 1st power.

[Much more will be added here.]

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