Calculus: Integrals of Trig and Exponent Functions

Great overview of derivatives!

06/01/2009

~ brittanie

In school it was really hard for me to learn how to find integrals or derivatives. Trig is hard! Especially when you're working with derivatives and trig functions. I like Professor Burger's teaching style. He's really easy to follow.

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The Basics of Integration
Antiderivatives
Antiderivatives of Trigonometric and Exponential Functions Page [1 of 1]
All right, so now we actually can find the integral, or the anti-derivative, of functions in the form xpn. We see that x equals the xn+1 divided by n+1 as long as n is not –1 and if n is –1, then we’re looking at 1/x and we know that the n is a derivative of a natural log of the absolute value of x because the derivative of a natural log of x is 1/x. But what about more interesting functions? Some of the exotic functions that we’ve seen when we were looking at the derivatives. Well, in fact, we can give a lot of those formulas immediately because of all the extra hard work we’ve done. For example, let’s think about Trig for a sec. Well, what is the integral of cos of xdx? So, the integral cos xdx, well, then I think to myself, what’s a function whose derivative is cos? If you think a little bit, I hope you’ll convince yourself that it’s sin, the derivative sin we’ve already established with cos, but don’t forget we have to add the plus a constant.
All right, how about the integral of sin? Well, I think a great guess would be to say cos, but a check would reveal that’s wrong because if I take the derivative of this, I’m suppose to get that. But the derivative of this, you may remember is negative sin, so actually I’m off by a negative here. So if I put in an extra negative, now when I take the derivative, I see negative negative sin, which is sin. So, in fact the anti-derivative or the integral of sin xdx is equal to minus cos x, and don’t forget we have to have the constant.
How about this one? How about the integral of sec2xdx. That sounds a little bit threatening. Do you think back and remember that actually we saw that the derivative of tangent was in fact sec2? So, in fact this is tangent x, but don’t forget the constant. You really think, gee, we skipped over the one that I would have thought of which would have been, let’s see, integral of tangent. So, let’s write that down as a question. What’s the integral of tangent? Well, we would need to think of a function, that when you take a derivative of it, you get tangent. Have we seen a function like that? Well, actually I don’t think we have. Isn’t that amazing? We’ve seen so many functions in this journey. How is it possible that we haven’t seen a function whose derivative is that simple thing? Well, the answer is we haven’t, but it turns out we will be able to figure this out in just a few more discussions. So, there’s a great wall of foreshadowing, maybe you want to think about it for a second. You maybe can’t believe—wait a minute, is that? Here’s a great guess, maybe it’s sec2? But sec2 is not the right answer, and all you have to do is take the derivative of sec2 and see you still don’t get tangent. You’ll get something much more complicated. Remember, also take the derivative of sec2, you’ll have to actually use a chain rule and you’ll get something really complicated. Try it if you don’t believe me and I think that would be great.
Any case, we do know the integrals of these things: cos, sin, and sec2. How about some other functions that aren’t trigonometric functions? For example, how about the integral of exdx? What’s the function whose derivative is ex? When you think about this, this is one of the wonderful things that we love about the function ex. It’s derivative is itself, and so therefore, it’s integral is itself. Why? Because the derivative of ex is ex. So, great, so exponential function ex is a great function to integrate or to differentiate, really ease and don’t forget the +c.
So, there are some more elaborate functions that we might want to consider taking the derivatives of. By the way, you’ll notice that I didn’t put in the integral of the log functions, just like I didn’t put in the integral of the tangent function. What’s the anti-derivative of the natural log? That asks the question what function has the property of its derivative is a natural log? We have all the other functions, the exponential function, how come that’s a natural log? Well, the answer is that we had not seen this yet either. And to fully understand this question, you know what you’ve got to do? You’ve got to take the next semester of calculus. Isn’t that neat? Already, you can start seeing the kind of things, the borders of knowledge are already making themselves known to us, and we’re just starting to work at this anti-differentiation, where you can see there is some very simple functions that we don’t even know how to find the anti-derivatives of. In some sense, finding anti-derivatives is actually more challenging than finding derivatives; that’s actually true. And we’ll see there are a lot of techniques used to find the anti-derivatives. Anyway, let’s leave these tantalizing questions for now, and definitely come back to this one and maybe, I’ll even give you a hint as to this one.
But instead, I thought we’d take some look at some simpler examples and work through a couple problems together. So, let’s work two problems together, the first one is finding the integral or the anti-derivative of 6x2 – 4x –1/x + 1dx. That’s a really long function and I want to find the anti-derivative. I want to find the function whose derivative is that. Okay, how do I proceed? Well, by the fact that we mentioned in the last discussion, I know that I can just break this up and this in this integral and then subtract it from this integral, subtract it from that integral, and then add it to that integral. Just like when we take derivatives, we can just take them piece by piece when we’re adding and subtracting. So, I’m going to use that fact right now, and I’m going to take the integral of each of these pieces individually. How do I proceed? Well, that constant, that 6 is a constant, I’m going to keep it out in front and then I now see x2, so what I do I add one to the exponent and I divide by that. So, I see x3/3. Then I subtract all four, and then I have x2, and then I divide by the two, and then I have a minus. Now, what do I do here? Well, now that method doesn’t work because that’s actually x-1, and so, what I see here is something over zero, but then I remember, aha, what we have to do is look at the natural log of the absolute value of x, and then what’s the derivative of one? Well, the derivative of 1 is just x right? Because the anti-derivative of 1 is x because the derivative of x is equal to one. And don’t forget we want the constant and so this equal if we simplify this, we see a 2x3 - 2x2 - the natural log of the absolute value of x + x + c. And that turns out to be the answer and you can check that answer real easy. How? Let’s take the derivative of that and see if we get back this. I’ll do it for you right now verbally. If I take the derivative of this, I see 3 x 2, which is 6x2, great. The derivative of this would be –4x, the derivative of the natural log is 1/x and the derivative of x is 1, the derivative constant is zero. This actually checks. This is really wonderful stuff, once you get them, because you can always check and know for sure if you’re right, there is no mystery here.
Great, okay, let’s do one last one together. Let’s find the integral of sin x + ex – (the cube root of x)2dx. Remember, this is just a symbol that captures the function that I want to take the derivative, find the anti-derivative of, find the integral of. Again, I’m going to take it piece by piece and see if I can figure this out. The anti-derivative of sin, well, that is minus cos plus the anti-derivative of ex is itself, ex minus… Now, what’s the anti-derivative of that? Well, that’s going to require us to do a little bit of work, I’m going to show you what I would do, personally, I would actually write that out so I have the whole thing as one big exponent. I remember that the cube root of x is the same things as x1/3. So in fact, the cube root of x all squared equals (x1/3)2 and that equals x2/3. Now, if I want to take the integral of that, what would I do? Well, I add one to the exponents. So if I add one to the exponent, I would see x2/3 + 1 which is 2/3 plus 3/3 which is 5/3. But, then I have to divide by 5/3 and so I divide by 5/3, and if you invert, whoop, and multiply, we see 3/5x5/3, and so that must be the integral here 3/5x5/3 + c, don’t forget the c. We can check our answer really fast, the derivative of cos is minus sin, put that minus in front of that to make sure a minus, minus which is a plus sin, the derivative of ex is ex, and what’s the derivative of this messy thing? Well, I bring the 5/3 in front, now watch this now, when I bring the 5/3 in front, the 5’s and the 3’s all cancel and I’m left with x raised to the 5/3 – 1 power. Well, 5/3 –1 is, in fact, 2/3, which is what we had. The derivative of constant zero, this answer is correct, we just checked it.
Okay, well, up next I want to start taking a look at some applications, why would we actually want to find anti-derivatives and what we are going to do to see that is to go back to the notion of velocity, go back to the notion of acceleration, and start seeing how movement can help us look at the meaning of this. I’ll see you there, bye.