Video: Finding the Integration of a Function Involving Using Polynomial Division

Determine ∫(4𝑥³ + 𝑥 + 1)/(2𝑥 + 1) d𝑥.

04:18

Video Transcript

Determine the integral of four 𝑥 cubed plus 𝑥 plus one over two 𝑥 plus one d𝑥.

Now, with this kind of problem, what we want to do first is we want to carry out something that’s known as a polynomial division or polynomial long division. So as you can see, the first thing I’ve done is set up a polynomial division. The key thing here is that I included any of our powers of 𝑥 that we haven’t got. So, for example, if we look, we’ve got four 𝑥 cubed plus 𝑥 plus one. We didn’t have any 𝑥 squareds. However, we include zero 𝑥 squared because it keeps things tidy when we’re doing a polynomial long division.

So the first thing we do is we divide four 𝑥 cubed by two 𝑥. And what we get is two 𝑥 squared. So we write this on top. Then next, we multiply two 𝑥 by two 𝑥 squared. Well, we already know what the result is gonna be. The result is gonna be four 𝑥 cubed. So we write this beneath the four 𝑥 cubed. Then next, what we do is we multiply positive one by two 𝑥 squared. So that’s gonna give us positive two 𝑥 squared. So we write this below the corresponding 𝑥 squared term. Okay, great. So now, what’s the next step?

Well, what we do is we subtract our terms. So first of all, we’re gonna have four 𝑥 cubed minus four 𝑥 cubed, which is just gonna give us zero. Then, we’ve got zero 𝑥 squared minus two 𝑥 squared, which is gonna give us negative two 𝑥 squared. So great, this is this step completed. So now, what’s the next step? So the first thing we need to do is bring down our 𝑥 term. So I’ll will put that down. So we’ve got negative two 𝑥 squared plus 𝑥. So now, what I want to do is divide negative two 𝑥 squared by two 𝑥, which is gonna give me negative 𝑥. And that’s because negative 𝑥 multiplied by two 𝑥 gives us negative two 𝑥 squared.

So again, we’re now gonna multiply two 𝑥 by negative 𝑥. Well, we’re already know that’s negative two 𝑥 squared. So again, I’ve written this beneath the same terms, so the 𝑥 squared term. So then next, what we need to do is I need to multiply positive one by negative 𝑥, which is gonna give us negative 𝑥. So I’ve written this below the positive 𝑥 term. So once again, the next stage is to subtract. So we’ve got negative two 𝑥 squared minus negative two 𝑥 squared. Well, this gives us zero. Then, we’ve got positive 𝑥 minus negative 𝑥, which is gonna give us positive two 𝑥.

So once again, what we do is we bring down the next term now, so our positive one. So we’ve got two 𝑥 plus one. And then, we’re gonna repeat once again for the final time. Well, two 𝑥 divided by two 𝑥 is just one. Then, two 𝑥 multiplied by one is two 𝑥. One multiplied by one is just one. So then, our final stage, which is our final round of subtraction, which is just gonna bring us zero because two 𝑥 minus two 𝑥 is zero. And one minus one is zero. So therefore, we know that four 𝑥 cubed plus 𝑥 plus one can be divided by two 𝑥 plus one without any remainder. And we know that the result of that polynomial division is two 𝑥 squared minus 𝑥 plus one.

Okay, great. So now we’ve divided and we simplified. What we can do is integrate this. So now, what we’re looking to do is integrate two 𝑥 squared minus 𝑥 plus one d𝑥. So our first term is gonna be two 𝑥 cubed over three. And we get that because what we do is we raise the exponent by one. So we add one to two to give us three. So that gives us two 𝑥 cubed. And then, we divide by the new exponent. So then, we’ve got minus 𝑥 squared over two plus 𝑥 and then plus our 𝑐, which is our constant of integration.

So therefore, we can say that the integral of four 𝑥 cubed plus 𝑥 plus one over two 𝑥 plus one d𝑥 is two 𝑥 cubed over three minus 𝑥 squared over two plus 𝑥 plus 𝑐.

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