WEBVTT 00:00:01.000 --> 00:00:07.885 As you know, there is no simple integration rule where products and quotients are concerned. 00:00:07.887 --> 00:00:13.500 The first thing you should always do when you have a product is try to multiply. When you have a quotient, 00:00:13.502 --> 00:00:20.000 try to divide. When that doesn't work, you have to be creative and use other techniques. Before we get to 00:00:20.002 --> 00:00:28.400 those techniques, we need to review some derivative rules. Let's take a look first at the general power 00:00:28.402 --> 00:00:42.001 power rule, or chain rule. The derivative with respect to x of u to the n, where u is an expression in x, 00:00:42.003 --> 00:00:54.700 it could be written u of x, is equal to n times u to the n minus 1 times the derivative of u 00:00:54.702 --> 00:01:01.801 with respect to x, and this is the part we want to pay careful attention to. Alright, now let's take the 00:01:01.803 --> 00:01:11.001 derivative of a function that has this form. So let's take the derivative with respect to x of, 00:01:11.003 --> 00:01:26.500 let's use something like 1/4 times 6 x squared minus 5x maybe to the fourth. So this is u to the n. 00:01:26.502 --> 00:01:37.900 Now we have a constant in the front here. So the derivative would be 4 times 1/4 is 1, so that would be 1 00:01:37.902 --> 00:01:52.401 times 6 x squared minus 5x, subtract 1, and then take the derivative of u. The derivative of u is 00:01:52.403 --> 00:02:04.299 12x minus 5, and this part right here is the du/dx. Start thinking of it and calling it that. Notice that 00:02:04.301 --> 00:02:12.500 it just appears. It isn't part of the original function. It just appears from the chain rule. 00:02:12.502 --> 00:02:25.199 So if we turn this around and we write this using the integral, this is what has to be in the integral 00:02:25.201 --> 00:02:33.301 in order for this term to be the antiderivative. So that would mean that the integral of 00:02:33.303 --> 00:02:59.200 6 x squared minus 5x cubed times 12x minus 5 dx equals this term, which is 1/4 (6 x squared minus 5x) 00:02:59.202 --> 00:03:06.901 raised to the fourth, and then of course we could have had a constant here. Now the classic question 00:03:06.903 --> 00:03:15.300 that I get is where did the 12x minus 5 go because it appears that it disappears. So what you want to 00:03:15.302 --> 00:03:25.699 remember here is that this term, the du/dx, came out so to speak of taking the derivative of u. 00:03:25.701 --> 00:03:35.001 So when you find the antiderivative it sort of goes back in. It has to be here. This term must be here 00:03:35.003 --> 00:03:42.700 for this to be the correct antiderivative, and that is the part that has to sink in on this technique. 00:03:42.702 --> 00:03:52.401 Now look at this term that's in here. This is a product. So at some point if you have a product, 00:03:52.403 --> 00:04:02.200 if you can recognize that this term here is the derivative of the inner part of this function, then you 00:04:02.202 --> 00:04:10.100 may be able to quickly with maybe a small adjustment find the antiderivative. Now this is a good example 00:04:10.102 --> 00:04:19.601 of how you use the general power rule. Let's look at the natural exponential rule. The derivative with 00:04:19.603 --> 00:04:36.601 respect to x of e to the u is e to the u times du/dx. So you see a similarity here, so let's do an example. 00:04:36.603 --> 00:04:53.200 The derivative with respect to x of e to the let's say 2x minus 1 is e to the 2x minus 1 times 2. 00:04:53.202 --> 00:05:04.401 And so what is this? This part represents the du/dx. So if we write the integral of, and I'm going to move 00:05:04.403 --> 00:05:12.195 this 2 to the front because that's the way you're going to see it most of the time, 2 e to the 2x minus 1 00:05:12.197 --> 00:05:25.200 dx, then the antiderivative is e to the 2x minus 1 with a possible c. So what happens here is the 2 00:05:25.202 --> 00:05:36.200 comes out from the chain rule, and the 2 goes back in. This part right here is the du/dx. It has to be 00:05:36.202 --> 00:05:43.001 here for this to be the correct antiderivative. This is the part again we're going to have to watch. 00:05:43.003 --> 00:05:48.901 Even though this is a constant, this is still a product, and it would be basically the same thing. 00:05:48.903 --> 00:05:56.401 If this was x squared, then we would have 4x here. Alright, let's take a look at a natural log. Write down 00:05:56.403 --> 00:06:08.701 the rule. The derivative with respect to x of the natural log of u, and again u is a function u of x. 00:06:08.703 --> 00:06:25.302 The derivative is 1 over u times du/dx. So let's take an example. The derivative with respect to x of 00:06:25.304 --> 00:06:47.901 the natural log of 4 x squared minus x is 1 over u times du/dx which would be 8x minus 1. And once again, 00:06:47.903 --> 00:06:57.901 what is this? This is the du/dx. It appears. So the integral of, what happens is when you see these 00:06:57.903 --> 00:07:06.100 in here, they don't come all spread out. You'll end up seeing this, and it'll look this. It looks like 00:07:06.102 --> 00:07:14.500 it's a quotient. It is a quotient when you put it in this way. So we have the integral of this term 00:07:14.502 --> 00:07:26.600 over this term and what we have here is the antiderivative has to be this. So it's the natural log of 00:07:26.602 --> 00:07:41.501 4 x squared minus x plus c. Of course we've got to be careful here because when we use this rule 00:07:41.503 --> 00:07:48.700 because we're going this direction the natural log function is not defined here. We actually have 00:07:48.702 --> 00:07:56.400 to put absolute value bars here because the natural log function has a restriction on the domain that 00:07:56.402 --> 00:08:02.101 says what's in this position has to be greater than 0. Since it didn't occur here, when we go backwards 00:08:02.103 --> 00:08:13.500 that has to go in. Now look at what this quotient looks like. It's actually 1 over u and then du/dx, and 00:08:13.502 --> 00:08:22.301 the du/dx disappears when you get the antiderivative, just like it appears here. Alright, now let's 00:08:22.303 --> 00:08:29.400 take a look at what these rules look like for integration. The first rule is used when you're finding 00:08:29.402 --> 00:08:37.901 the antiderivative of a function of the form u to the n times du/dx, except when n equals negative 1. 00:08:37.903 --> 00:08:43.301 The second rule is used to find the antiderivative when you have e raised to a functional power, 00:08:43.303 --> 00:08:50.898 so that would be e to the u times du/dx. The third rule is used to find antiderivative when you have 00:08:50.900 --> 00:08:56.800 a function in the denominator raised to the first power, so that would be 1 over u. Or you have an 00:08:56.802 --> 00:09:01.605 expression in the numerator raised to the negative 1 power, so that would be u to the negative 1 and 00:09:01.607 --> 00:09:08.900 then times du/dx. Note that the natural log has absolute value in it. That's because of the restriction 00:09:08.902 --> 00:09:14.001 on the domain of a log function requiring that the expression in that position be positive. 00:09:14.003 --> 00:09:21.800 Now for any of these problems, any problem using these three rules, you have to identify u; you have to 00:09:21.802 --> 00:09:28.800 take the du/dx; and then you have to be sure that it is correct. Also don't forget the c when you integrate. 00:09:28.802 --> 00:09:33.700 Now we're going to take a look at example one. I already have it set up for you with the correct u and the 00:09:33.702 --> 00:09:44.500 du/dx all identified. So the antiderivative will be u. See now all this is of course inside this integral. 00:09:44.502 --> 00:09:57.600 So the antiderivative would be x squared minus 1, which is the u. Then add 1, over 5, plus c. This step 00:09:57.602 --> 00:10:05.700 right here, and let's put this in here, this step is a thinking step. All I've done is rewrite this 00:10:05.702 --> 00:10:12.801 integral as, and you think, it's u to the fourth, that's the right du/dx, dx, and so the antiderivative 00:10:12.803 --> 00:10:20.000 is...poof, and it pops out. Now the question of course always is where did the 2x go, but you remember 00:10:20.002 --> 00:10:25.205 from what we did earlier that it came out when you took the derivative of this function in the first place because 00:10:25.20 --> 00:10:31.300 when you do the derivative of this the derivative of x squared minus 1 is 2x, and that's why it has to be there. 00:10:31.300 --> 00:10:39.313 So let's circle this. And can you check this? Of course you can. Take the derivative of this term. 00:10:39.315 --> 00:10:46.312 It would be 5 over 5, which cancels, x squared minus 1 to the fourth times 2x, which is exactly what we have 00:10:46.314 --> 00:10:53.609 right here. Ok, this little worksheet, it's probably a good idea to put it in some sort of little container. 00:10:53.611 --> 00:11:14.212 Example two. The integral of 5 x to the fourth plus 2 over x to the fifth plus 2x and then that 00:11:14.214 --> 00:11:24.408 quantity squared dx. Alright. This is a quotient. There is no quotient rule, and so we look to see if we 00:11:24.410 --> 00:11:30.009 can divide. Well, obviously we're not going to be able to divide this and get anything simple. So now we 00:11:30.011 --> 00:11:38.109 look for a match possibly at one of these rules. There is no e to the x at all so that leaves this rule 00:11:38.111 --> 00:11:47.314 out completely. Because this is a quotient, you may at first be drawn to this rule, but if you look at 00:11:47.316 --> 00:11:58.812 this, it says u. There's no n here. This is u to the one. Our problem is u squared, so this rule is out. 00:11:58.814 --> 00:12:04.808 So it's either this one, or at this point nothing unless you know other techniques. So let's see if we 00:12:04.810 --> 00:12:23.007 can rewrite this and make it match this pattern. We could rewrite x to the fifth plus 2x to the negative 2. 00:12:23.009 --> 00:12:38.808 So now u is x to the fifth plus 2x. Negative 2 is the n. Then the numerator here, 5 x to the fourth plus 2, 00:12:38.808 --> 00:12:54.210 may be the correct du/dx, and if it is, we're ready to roll. Let's check it. u is, we think, x to the fifth plus 2x. 00:12:54.212 --> 00:13:05.411 Then the correct derivative, du/dx, is 5 x to the fourth plus 2. So again, we're all happy because this 00:13:05.413 --> 00:13:20.810 is the correct du/dx. This rule applies immediately, and so the antiderivative is, this is continuing 00:13:20.812 --> 00:13:34.312 from here. The antiderivative is u, and the rule says add 1, so we have negative 2 plus 1 is negative 1 00:13:34.314 --> 00:13:44.809 divided by negative 1 plus c. Now you may have noticed that when I do this little work tablet here, 00:13:44.811 --> 00:13:53.206 I say the correct du/dx is. I'm leading up to the possibility that the term here may not be exactly the 00:13:53.208 --> 00:14:01.212 correct du/dx, but it might be close enough that we can make some modification and still match the pattern. 00:14:01.214 --> 00:14:19.012 Let's go to example three. The integral of x times 4 x squared minus 3 raised to the, let's try a 00:14:19.014 --> 00:14:28.910 fraction this time, 1/5 dx. Alright, there's no e. There is no term in the denominator, so we're trying to 00:14:28.912 --> 00:14:35.613 match this one, and of course this is the most common one. So let's see what we can do. We could think of this as u. 00:14:35.615 --> 00:14:41.312 This is n, and we need to see if this is the correct du/dx. I'm sure you can look at it and tell that 00:14:41.314 --> 00:14:56.209 it is not, but let's write it down. This is 4 x squared minus 3, and so the correct du/dx would be 8x. 00:14:56.211 --> 00:15:06.412 Well, this is not a match exactly, but the variable is correct, the x, and the only thing we really need 00:15:06.414 --> 00:15:15.510 is an 8. Now if you remember one of the format rules we had, way back when we were working earlier with 00:15:15.512 --> 00:15:22.508 the first integration in an earlier topic. We had a rule that says a constant in here could be pulled out 00:15:22.510 --> 00:15:30.213 of here or put back in here. So what we do is, we do what's called adjusting the constant. We want an 8 in here 00:15:30.215 --> 00:15:38.410 so that this is the correct du/dx. You've probably always wanted to do this, so you just put in an 8. But 00:15:38.412 --> 00:15:47.212 of course you've changed the problem, so on the outside you multiply by 1 over 8. Now you have to squeeze 00:15:47.214 --> 00:15:57.209 it in here, but it works. Because what have we done? We've added to the problem this, 1/8 times 8 which is 1. 00:15:57.211 --> 00:16:06.012 So we have changed nothing, except that this is now the correct du/dx. It matches this pattern. 00:16:06.014 --> 00:16:14.318 So we're ready to get the antiderivative. Now don't lose the 1/8. It is still on the outside of the 00:16:14.320 --> 00:16:32.509 integral, so when we find the antiderivative, it's u to the n plus 1, so this would be 1/5 plus 5/5 is 6/5. 00:16:32.511 --> 00:16:42.913 Then we have to divide by 6/5, but that's the same thing as times 5/6, and then we have plus c. 00:16:42.915 --> 00:16:51.611 Now this gets a little messy in here. You could rewrite this as, and let's see. We could rewrite this as 00:16:51.613 --> 00:17:01.513 just 1/8 times 5 over 6, times... oh, how'd that get to be a 3? That should be a 2. See we always go back 00:17:01.515 --> 00:17:12.306 and check these. So then this would be 4 x squared minus 3 to the 6/5 plus c. Alright, let's go back and 00:17:12.308 --> 00:17:25.408 check over the whole thing. u to the n. The derivative of u is 8x. We put in an 8. We multiply by 1/8. 00:17:25.410 --> 00:17:35.314 So the product is 1. The 1/8 stays on the outside. This is u to the n plus 1, and let's make sure we copy 00:17:35.316 --> 00:17:44.912 u exactly. Then it's divided by 6/5 or times 5/6. Now of course we could multiply this and put 5 and then 00:17:44.914 --> 00:17:50.209 the whole thing over 48, but it depends on what you want to do with this whether that is necessary. 00:17:50.211 --> 00:18:01.012 At this point, we're satisfied to just practice the skill. Let's do another one. Example four. 00:18:01.014 --> 00:18:22.610 So suppose we have the integral of x squared plus 1 over x cubed plus 3x to the fourth dx. 00:18:22.612 --> 00:18:29.412 Very similar problem. This one is out, and this rule is out because this is to the fourth not to the one. 00:18:29.414 --> 00:18:42.012 So we're going to try to use this rule. So we're going to rewrite this as the integral of x cubed plus 3x 00:18:42.014 --> 00:18:55.615 raised to the negative 4 and then times x squared plus 1 dx. Ok, if you're looking ahead here, you 00:18:55.617 --> 00:19:19.711 can see that u is going to be this term. But what's the correct du/dx? It's 3 x squared plus 3, which is 00:19:19.713 --> 00:19:29.311 not a match. But what's wrong? Well the only thing that's wrong here is that 3 times x squared plus 1 00:19:29.313 --> 00:19:35.313 is equivalent to this term. We have the x squared plus 1. All we need is the 3. So we're going to do the 00:19:35.315 --> 00:19:43.409 same thing. Now this looks messy when you first do it. Most people are happiest if they put the 3 right there. 00:19:43.411 --> 00:19:51.111 Now what that would mean is we'd have to have a 1/3 out here. So let's recopy that so you can actually see it. 00:19:51.113 --> 00:20:03.911 So what we now have is x cubed plus 3x raised to the negative 4 times 3 times x squared plus 1 dx. 00:20:03.913 --> 00:20:13.111 Alright, let's stop and look at it. This is u to the n. This is the correct du/dx, so now we have this product. 00:20:13.113 --> 00:20:29.811 We can get the antiderivative quickly. It is 1/3 times u to the n plus 1, so that would be negative 3, 00:20:29.813 --> 00:20:42.511 all divided by negative 3 plus c. Now this is a messy problem when you write it all out. It's a good time to 00:20:42.513 --> 00:20:48.910 maybe use a circle. Of course you should check this. You can take the derivative of this function, and you 00:20:48.912 --> 00:20:55.308 should eventually be able to simplify that derivative and get back to this. You could multiply this, put 00:20:55.310 --> 00:21:03.116 negative on the outside, 9 down here, but we stop at this point. This is called adjusting the constant. 00:21:03.118 --> 00:21:26.013 Alright, let's take a look at more examples. Number five. The integral of 4 times 2 x squared minus 7 to the 00:21:26.015 --> 00:21:35.008 third power and then dx. Ok, this one's out. This one's out. We're going to try to match this. 00:21:35.010 --> 00:21:39.610 So it looks like we have u to the third. So let's do what we know. What we're going to do now, 00:21:39.612 --> 00:21:49.712 we're going to take u. Write it down. We're going to take the derivative of u. And what do we get? 4x. 00:21:49.714 --> 00:21:59.911 Oops. We have a problem. The 4 is fine, but we need the x. Avoid the temptation of putting in the x and 00:21:59.913 --> 00:22:06.410 multiplying by 1 over x. You can't do that. You will not get the right antiderivative. There is no rule 00:22:06.412 --> 00:22:11.508 that allows you to float a variable in and out. Obviously a variable in here has a totally different 00:22:11.510 --> 00:22:19.810 meaning out here. So at this point, what would we do? Well, I think we'd have to cube this, distribute 00:22:19.812 --> 00:22:26.608 the 4, write it as a sum and difference formula, and take the integral in that direction. I think we will 00:22:26.610 --> 00:22:47.410 leave that and go on to example 6. The integral of e to the x over e to the x minus 1 to the fifth dx. 00:22:47.412 --> 00:22:54.913 I picked this example here because I think you need to see what's going to happen when you try to match 00:22:54.915 --> 00:23:03.309 a pattern. As soon as you see an e in this problem, you are drawn to this rule, but look at the rule. 00:23:03.311 --> 00:23:14.810 It says e to the u, and then there's the derivative of u. Well this could be e the the u. But then what's this? 00:23:14.812 --> 00:23:22.911 So what you really have here is you have an exponential function inside of this u, and this is the pattern 00:23:22.913 --> 00:23:33.013 we're actually going to follow. Let's rewrite it. This would be e to the x minus 1 to the negative 5 00:23:33.015 --> 00:23:48.009 times e to the x dx. This is u. u is e to the x minus 1, and the derivative of that is e to the x, and 00:23:48.011 --> 00:23:56.208 it's here. Correct. No adjustment necessary. We have u to the negative 5. It's this pattern even though 00:23:56.210 --> 00:24:08.410 there's an e in here. The antiderviative would be e to the x minus 1, add 1, divide by that number, 00:24:08.412 --> 00:24:18.612 plus c. So just because there's an e in here somewhere doesn't mean it's going to match this pattern. 00:24:18.614 --> 00:24:25.210 If it's an e inside a function with some other terms raised to a power, the match you usually go for is 00:24:25.212 --> 00:24:45.009 this one. Alright, example seven. The integral of 2 e to the 2x minus 1 times e to the 2x minus x 00:24:45.011 --> 00:24:55.309 to the seventh dx. Ok, when you first look at this problem, it actually looks pretty horrible, but once again, 00:24:55.311 --> 00:25:01.408 it is not this rule because it's not of the form e to the u. It's actually u to the n. This is u. 00:25:01.410 --> 00:25:17.905 This is n. Is this the correct du/dx? Let's check it. du/dx would be e to the 2x times 2 minus 1. 00:25:17.907 --> 00:25:28.909 Is that exactly what we have here? Well, this is 2 e to the 2x minus 1. That is an exact match. 00:25:28.911 --> 00:25:45.008 Antiderivative is u to the n plus 1 over 8 plus c, and now that you're getting better at this, we can 00:25:45.010 --> 00:26:02.508 pick up the pace just a bit. Let's look at example eight. The integral of 1 over x times ln x squared dx. 00:26:02.510 --> 00:26:07.909 Now before we work on this, let's stop and take a moment to appreciate what you have actually learned 00:26:07.910 --> 00:26:16.509 how to do here. You know how to find the antiderivative of a term that looks like this or a term that 00:26:16.511 --> 00:26:26.708 looks like this simply because you can make it match a known pattern. Can we make this match? Well, 00:26:26.710 --> 00:26:36.209 what's the derivative of ln x? It's 1 over x. This is ln x raised to a power. Do not be drawn to this. 00:26:36.211 --> 00:26:42.613 Notice that here the antiderivative is ln x. This has nothing to do with the natural log being in here. 00:26:42.615 --> 00:26:52.110 So let's rewrite it as something we can work with. This would be ln x quantity squared but 00:26:52.112 --> 00:27:01.609 raised to the negative 2 in the numerator, and then this is times 1 over x. Can you see what we have? 00:27:01.611 --> 00:27:15.010 We have u to the n. We have the correct du/dx. Antiderivative is natural log of x plus 1 over the power 00:27:15.012 --> 00:27:21.709 plus c. You can do a worksheet on this if you want to, but at some point you get pretty good at mentally 00:27:21.711 --> 00:27:36.009 checking this is the correct du/dx, and I think we're getting to that point. Number nine. This particular 00:27:36.101 --> 00:27:51.206 example is one I always put on a test. It's really a way to determine if you can actually manipulate this 00:27:51.208 --> 00:27:56.911 and see what's really happening. Well, what's really happening is that someone has taken the derivative 00:27:56.913 --> 00:28:07.008 and simplified it, and this is the simplified version. What you really have here is the natural log of x 00:28:07.010 --> 00:28:19.508 raised to the first power, and see that makes this one a lot like eight. And then times 1 over x dx. This 00:28:19.510 --> 00:28:26.308 doesn't come to you in parentheses. It's a little harder to recognize it as u to the n, but it's actually 00:28:26.310 --> 00:28:35.005 u to the 1. So the antiderivative, let's check it. This is u. This is the correct du/dx. So the 00:28:35.007 --> 00:28:52.709 antiderivative is u, add 1, that's a 2, over 2, plus c, and again it's the first rule. Example ten. 00:28:52.711 --> 00:29:13.508 The integral of x squared e to the x cubed minus 4 dx. Finally, e to the u. Is this the correct du/dx? 00:29:13.510 --> 00:29:34.110 u is x cubed minus 4. The derivative is 3 x squared. What can we do? We need a 3. We put a 3 here, 00:29:34.112 --> 00:29:43.012 a 1/3 on the outside. Now this is the correct du/dx, and we can follow this rule which is, watch the 1/3, 00:29:43.014 --> 00:29:53.810 it stays with you, and then the antiderivative is e to the u. Avoid the temptation to add 1, 00:29:53.812 --> 00:30:14.912 and then put plus c. A few more. The integral of e to the negative 1 over x all over x squared dx. 00:30:14.914 --> 00:30:22.010 Now if you're getting some negative feelings from this problem, that's ok. It does look a little intimidating, 00:30:22.012 --> 00:30:28.409 but let's follow the pattern we've done. It's obviously e to the u, but then there's this other part. 00:30:28.411 --> 00:30:35.009 So separate it because you see someone has just simplified a derivative, so we're going to unsimplify it. 00:30:35.011 --> 00:30:46.813 This is e to the negative 1 over x, and then we have left times 1 over x squared dx. 00:30:46.815 --> 00:31:01.011 u is the power of e which if you recall of course is x to the negative 1. So the du/dx would be 00:31:01.013 --> 00:31:12.613 positive x to the negative 2, and look what it is. It's 1 over x squared, and that's exactly what we have here. 00:31:12.615 --> 00:31:26.409 So the antiderivative of this nasty looking original expression is e to the u plus c. Once again, 00:31:26.411 --> 00:31:37.308 that's because the correct du/dx is here. The next two problems are slightly different. The integral of 00:31:37.310 --> 00:31:52.108 x minus 2 over x squared minus 4x dx. Why is this different? Well, first of all u is not so obvious, 00:31:52.110 --> 00:31:57.509 but again we realize that this is x squared and this is x, so the numerator is more likely to be the 00:31:57.511 --> 00:32:07.108 derivative of the denominator than the reverse. And notice that this term, if we think of u and identify 00:32:07.110 --> 00:32:18.408 it as x squared minus 4x, what's the power in the denominator? It's this term raised to the 1. 00:32:18.410 --> 00:32:28.112 So we separate. Well, let's check the du/dx while we're right here. It should be 2x minus 4. It differs 00:32:28.114 --> 00:32:39.810 only by a constant 2. We need a 2. We put in a 2. Multiply by a half, and so we have, let's rewrite this one. 00:32:39.812 --> 00:32:59.609 We have 1/2 times the integral of 1 over x squared minus 4x times 2 (x minus 2) dx. 1 over u. 00:32:59.609 --> 00:33:19.010 du/dx. Correct, and then dx, and so the antiderivative is the natural log of the absolute value of u plus c. 00:33:19.012 --> 00:33:29.712 And you see it's all about recognizing that this product is a result of taking the derivative. Now what 00:33:29.714 --> 00:33:34.709 would happen that's interesting is if you check this and you took the derivative of this, you would 00:33:34.711 --> 00:33:40.610 discover that you left out something very interesting, and this is very common. This little constant 00:33:40.612 --> 00:33:49.811 hanging around in the front has to be here. So I would encourage you on problems like this to check. 00:33:49.813 --> 00:33:56.812 Not only just to look back, but to actually take the derivative. Let's do one more. 00:33:56.812 --> 00:34:19.113 The integral of e to the x over e to the x minus 1 dx. Ok, let's analyze what we have here. We have e. 00:34:19.115 --> 00:34:32.713 Does it match this pattern? It's not e to the u. This could be e to the u, but then you've got the minus 1, 00:34:32.715 --> 00:34:40.213 so it doesn't match this one. Now you might think, well, ok, could we divide? No, that doesn't work. 00:34:40.215 --> 00:34:49.411 Does it match this one or this one? This term here in the denominator, you can write this in, is raised 00:34:49.413 --> 00:34:57.913 to the first power. So even though there's an e in here, it's actually going to be this third rule. 00:34:57.915 --> 00:35:12.708 So what do we have? u is e to the x minus 1. The du/dx is e to the x. We got a match. So we have here 00:35:12.710 --> 00:35:23.511 the integral of 1 over the denominator, that's u. Let's put the 1, times e to the x, which is the 00:35:23.513 --> 00:35:32.410 correct derivative of u, dx. So the antiderivative using the third rule is the natural log of the 00:35:32.412 --> 00:35:46.810 absolute value of e to the x minus 1 plus c. There are many other techniques that allow you to 00:35:46.812 --> 00:35:53.111 integrate using products and quotients. This is the first one, and this is the one that is most 00:35:53.113 --> 00:36:01.613 commonly used in applied calculus applications. This skill takes a lot of practice. 00:36:01.615 --> 00:36:07.000 Practice this before you go on to the next topic.