Use partial fractions to evaluate the integral of 16𝑥 cubed over four 𝑥 squared minus four 𝑥 plus one with respect to 𝑥.
Now, we are actually told how we need to solve this problem. We need to write 16𝑥 cubed over four 𝑥 squared minus four 𝑥 plus one as partial fractions. Before we can though, we need to notice that the highest power in the numerator is greater than the highest power of 𝑥 in the denominator. And so, this is a top-heavy fraction. We’re going to need to use polynomial long division before we can actually use partial fraction decomposition. And so, we’re going to divide 16𝑥 cubed by four 𝑥 squared minus four 𝑥 plus one. Now, let’s write 16𝑥 cubed as 16𝑥 cubed plus zero 𝑥 squared plus zero 𝑥 plus zero. This just makes the following steps a little easier.
The first thing we do then is divide 16𝑥 cubed by four 𝑥 squared. That’s four 𝑥. Then, we individually multiply four 𝑥 by four 𝑥 squared, negative four 𝑥, and one. That gives us 16𝑥 cubed minus 16𝑥 squared plus four 𝑥. The next thing we do is we subtract each of these terms. 16𝑥 cubed minus 16𝑥 cubed is zero. Zero 𝑥 squared minus negative 16𝑥 squared is 16𝑥 squared. And zero 𝑥 minus four 𝑥 is negative four 𝑥. Let’s bring down this zero. Next, we divide 16𝑥 squared by four 𝑥 squared, which is simply four. And then, we multiply this four by each term in the denominator. That gives us 16𝑥 squared minus 16𝑥 plus four. And then we subtract one last time. And we’re left with 12𝑥 minus four.
Now, what this actually means is we can write 16𝑥 cubed over four 𝑥 squared minus four 𝑥 plus one as four 𝑥 plus four plus 12𝑥 minus four all over four 𝑥 squared minus four 𝑥 plus one. Now, this part here is what we’re actually going to write in partial fraction form. Before we can though, we need to ensure that the denominator is fully factored. And when we factor four 𝑥 squared minus four 𝑥 plus one, we get two 𝑥 minus one all squared. And now, we can write our fraction as the sum of two rational functions. It’s 𝐴 over two 𝑥 minus one plus two 𝐵 over two 𝑥 minus one all squared, where 𝐴 and 𝐵 are constants.
We’re going to add the fractions on the right-hand side by creating a common denominator of two 𝑥 minus one squared. To achieve this, we multiply the numerator and denominator of our first fraction by two 𝑥 minus one. And so, we get 𝐴 times two 𝑥 minus one over two 𝑥 minus one squared plus 𝐵 over two 𝑥 minus one squared. And since the denominators are equal, this is 𝐴 times two 𝑥 minus one plus 𝐵 over two 𝑥 minus one squared. Now of course, these two fractions are equal. And we can see now that their denominators are equal. So, this must mean their numerators themselves are equal.
In other words, 12𝑥 minus four must be equal to 𝐴 times two 𝑥 minus one plus 𝐵. And we have a couple of ways that we can solve this. We can substitute in roots or we can distribute the parentheses and acquire coefficients. We’re going to use the latter method. When we distribute the parentheses on our right-hand side, we get two 𝐴𝑥 minus 𝐴 plus 𝐵. We begin by equating coefficients of 𝑥 or 𝑥 to the power of one. On the left-hand side, that’s 12. And on the right, it’s two 𝐴. We divide both sides of this by two. And we find 𝐴 to be equal to six.
We do this again with 𝑥 to the power of zero, the coefficients of 𝑥 to the power of zero. But they’re just the constant terms. So, on the left-hand side, we have negative four. And on the right, it’s negative 𝐴 plus 𝐵. But of course, 𝐴 is six. So, we find negative four equals negative six plus 𝐵. And if we add six to both sides, we find 𝐵 to be equal to two. And so, this means 12𝑥 minus four over two 𝑥 minus one all squared is equal to six over two 𝑥 minus one plus two over two 𝑥 minus one squared.
We’re going to clear some space and now perform the integral. It’s the integral of four 𝑥 plus four plus six over two 𝑥 minus one plus two over two 𝑥 minus one squared with respect to 𝑥. We’ll do this term by term. When we integrate four 𝑥, we add one to the exponent and divide by that value. So, we get four 𝑥 squared over two which is two 𝑥 squared. The integral of four is four 𝑥. And then, we know that when we integrate one over 𝑎𝑥 plus 𝑏, where 𝑎 and 𝑏 are constants, we get one over 𝑎 times the natural log of 𝑎𝑥 plus 𝑏. And so, six over two 𝑥 minus one integrates to six times a half times the natural log of the absolute value of two 𝑥 minus one, which is three times the natural log of the absolute value of two 𝑥 minus one.
We do still need to integrate two over two 𝑥 minus one all squared though. And so, we’re going to perform a substitution. We’re going to let 𝑢 be equal to two 𝑥 minus one. This means that d𝑢 by d𝑥 is equal to two. Now, whilst d𝑢 by d𝑥 isn’t a fraction, we do treat it little like one. And we can say that d𝑢 must be equal to two d𝑥. And so, by replacing two d𝑥 with d𝑢 and two 𝑥 minus one with 𝑢, we see that we actually need to integrate one over 𝑢 squared with respect to 𝑢 or the integral of 𝑢 to the power of negative two with respect to 𝑢. Well, that’s 𝑢 to the power of negative one divided by negative one or simply negative 𝑢 to the power of negative one. Now, we’re going to replace this 𝑢 with two 𝑥 minus one. And of course, we must add our constant of integration 𝑘.
And so, we’ve used partial fractions to evaluate our integral. And we get two 𝑥 squared plus four 𝑥 plus three times the natural log of the absolute value of two 𝑥 minus one minus two 𝑥 minus one to the power of negative one plus some constant of integration 𝑘.