How to Choose the Right Integration Technique (u-Sub vs. Integration by Parts)

[00:00:00]These three integrals look almost identical, but each one requires a completely different strategy. Can you figure out which integration technique to use before you start solving?

[00:00:10]Welcome to Math with Professor V. In this video, I'm going to walk you through three very similar-looking integrals that each require a different approach. And by the end of the video, you'll have a better grasp and know what clues to look for so you can choose the right integration method instead of just guessing. And if you're looking for a more structured way to master calculus, stick around where I'll tell you about my online courses. In this first problem, we have indefinite integral of x * sin of x² dx. The integration technique that we're going to need to utilize here is u-substitution.

[00:00:46]What's the big clue? Notice we have a composition of functions. We have sin of x² and we have the derivative of the inside function multiplied outside. Now, we know that the derivative of x² is 2x and we don't have exactly that sitting outside. We just have 1x, but that's good enough for our purposes. If we're ever off by a constant, that's totally fine. We can adjust. So, let me show you how. What we start off by doing is we say, "Okay, we're going to let u = x² and then you differentiate both sides.

[00:01:21]So, du is equal to 2x dx. And then you go on a hunt and you go, "Ooh, do I see this anywhere in my integral?"

[00:01:30]Almost. Almost. I see x dx. So, what I tell my students to do when they're first learning u-sub is draw your little deformed kidney bean around what du resemblance you have.

[00:01:43]Only x dx in this case, so let me divide by two on both sides so it matches perfectly. So, 1/2 du is equal to x dx.

[00:01:52]And then now we're pretty much ready to go. So, I'm going to replace the kidney bean, x dx with 1/2 du, and then what's left is sine of x squared, but remember x squared is u. So, you want to change your whole integral from being in terms of x to being in terms of u in one step.

[00:02:10]You don't want to have mixing u's and x's. It's not well defined. It's not good. So, don't do it that way. Okay.

[00:02:17]So, we've got 1/2 du, but I'm going to put that at the end, right? You don't put it in the middle. And then we have sine of u. Boom.

[00:02:27]Are we okay so far?

[00:02:29]Okay, fantastic. Sine of u, and then this is 1/2 du. Always put the du on the right. Now, the constant I always encourage you to take it outside when you can. Okay, from here I think we're ready to go. Whose derivative did you take to get sine u?

[00:02:44]You would have had to take the derivative of a negative cosine u.

[00:02:47]That's right.

[00:02:49]And then now I already integrated, so I'm just going to put plus c. Get rid of the integral sign and no differential.

[00:02:55]So, this and this go away at the same time.

[00:02:59]And then what are we left with? Well, just go back, replace u with right here, x squared, what it originally was. So, this is going to be -1/2 cosine of x squared + c.

[00:03:13]Okay? And then you're done. How was that?

[00:03:17]Not bad, right? I love it so much. Okay, let's box it and move on with our lives. Next integral was indefinite integral x * sine of x dx. Now, this one we're going to need to use integration by parts. How do I know that? Well, I have a product of two functions, x and sine of x, and there's no useful composition. Notice the last one that we just did, we had sine of x squared, so that composition was useful because the derivative was sitting and being multiplied outside. In this case, the derivative of x is just one, so it's not going to help me. I still have this extra x out here. So, that's why in this sort of scenario, we're going to do integration by parts.

[00:03:56]Now, when you do integration by parts, remember part of the integrand you choose to be U and the rest is going to be DV. How do you choose them appropriately? I have several videos explaining. You can use an acronym, but I never do.

[00:04:09]I just set it up the way my calculus teacher taught me decades ago.

[00:04:13]So, when you set up U and DV this way, after that you find du and V, and you want this product to be as easy as possible to integrate. Just as simple as can be. So, that's why most of the time when you have a polynomial, you let it be U.

[00:04:29]So, I'm going to let U be X. That way when I differentiate and find du, it's just going to be 1 dx. And then the rest can be dv, all of this. So, dv is going to be sin x dx. So, to find V, you just integrate that in your head quickly and it's going to be negative cosine x.

[00:04:47]So, your formula, right?

[00:04:50]U dv, our original integral becomes uv minus integral v du. And just in my head when I set it up stacked this way, I always do this diagonal product outside.

[00:05:03]So, that's going to be negative x cosine x minus integral v du, which is here. Since I have another minus sign, that's going to make this plus cosine x dx. Okay? If you haven't seen integration by parts before and this was just too much, too fast, don't worry. I have so many video lectures in the description where I'll take it at a slower pace.

[00:05:28]Okay. And then from here, we should be home free, right? Negative x cosine x, we don't have to integrate that.

[00:05:34]Fantastic. And then antiderivative of cosine x?

[00:05:38]It's positive sine x. Good. Just say, whose derivative did I take to get cosine x? It's sine x.

[00:05:46]Badabing badaboom. Last example.

[00:05:49]Integral of x cubed times sine of x squared dx.

[00:05:53]This problem requires both integration techniques. So fancy. Notice we do have a composition of two functions, sine of x squared, and the derivative of x squared is 2x. We kind of have that, but we have more. We have an extra x squared plus more. So what I'm going to do is start with u sub, and then from there, I'm going to need to do integration by parts. I know that cuz I've done problems like this for decades, but you're probably going to just have to decide as you go.

[00:06:21]Now, since I know I'm going to use by parts after u sub, I want to save the variable u for the by parts moment. So I'm going to make a t sub. Look at me go. I'm going to let t equal x squared.

[00:06:35]And then differentiate, so dt equals 2x dx. Now if you try to draw your kidney bean, look what's going to happen.

[00:06:44]I don't have 2x dx, but what I'm going to write this as is x squared times x times sine of x squared dx. So okay, now I can see here 1/2 dt is x dx.

[00:06:59]Do you have a kidney bean of x dx? Yes, x dx is right here.

[00:07:04]And then what do I do with this extra x squared?

[00:07:08]Well, that's t.

[00:07:10]Ah, so this is going to be t.

[00:07:14]This x dx is 1/2 dt.

[00:07:20]Draw that there. And then you have sine of x squared, so that's going to be sine of t.

[00:07:27]All right, but don't put the dt squished all in the middle like that. Oh my goodness, it's suffering. So we're going to put the 1/2 outside actually. And then we'll have t sine of t dt. How we doing?

[00:07:41]Oh, perfect. Does this look familiar?

[00:07:43]Yes, it does. It's time to use integration by parts.

[00:07:47]And good thing I saved you and DV for this moment. So, U is going to be T, DV is going to be sin T dt, du is dt, and then find V, that's going to be negative cosine T.

[00:08:02]All the while this 1/2 is outside minding its business, behaving nicely.

[00:08:07]And then we're going to have U * V, so that's going to be - T cosine T - integral of V du, and then again, this minus sign will switch, so this is going to be plus cosine T dt.

[00:08:26]Perfect.

[00:08:28]And then just distribute and we can integrate at the same time. - 1/2 T cosine T.

[00:08:34]Antiderivative of cosine T is just positive sine T. Okay, perfect. So, plus 1/2 sine T + C.

[00:08:45]Now, before you get carried away and think, "Ooh, let me box this." Remember, our original variable was X, and all the way back here we had let T be X squared. So, let's go back and replace it accordingly. So, this is - 1/2 X squared cosine of X squared + 1/2 sine of X squared + C.

[00:09:09]Beautiful. Now, box that with pride.

[00:09:12]Good job, you guys.

[00:09:14]So, there you have it, a mix of different techniques for very similar looking integrals. And if you need more practice, if you're new to this material, you have come to the right place. I have over a thousand free video lectures here on YouTube and they're organized into playlists. So, if you need more help with your integration or calc two topics, go to the calculus two video lectures playlist. I will link it in the description, as well. And if you're taking calculus right now and looking for a complete course with guided notes, step-by-step video lessons, practice problems, sample exams, and lifetime access, be sure to check out my online courses at professurVmath.com.

[00:09:52]I'll put the link in the description below.

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