 Hi, I'm Zor. Welcome to Unizor education. I think it's time to do some exercises with indefinite integrals. We spend some time to define what basically is indefinite integral as an operation which is inverse to differentiation. I also explained some physical considerations which led to the concept of integral. Speed, distance, how to basically find out what's the distance, if you know the speed at every point. So let's just do some exercises. I have 10 different very very simple exercises on integration and let's just do them just one by one. Number one, what is integral of just gx? Well, I mean you might just stop and think about, usually we're talking about integral of f of x dx equals g of x plus c where derivative of g is equal to f, right? And they don't have an f of x here, right? There is no function. Well, there is a function. The function is equal to 1, obviously. So the function which is always equal to 1, that's my function, it's a constant. Now the question is, what is the function derivative of which is equal to 1? Well, obviously the function is function x. So function g of x is equal to x. If you take a derivative, that's x to the power of 1 and the derivative would be 1 times x to the power of 1 minus 1, which is 0. So we have 1. That's number one. Number two, I have integral of x to the power of adx. Now, I will write an answer just because I remember what is the derivative of the power function. So the power function, whenever I put something like this, I know that derivative reduces the power by 1. And there is a factor here, right? Derivative of this is a plus 1 times a plus 1 minus 1, which means x to the power of 8. So I've got x to the power of a, but I have this factor a plus 1, which we have to neutralize somehow. So to neutralize, I divide it by a plus 1 and then plus c. Obviously a should not be equal to minus 1 in this case. For minus 1, it's a different story. We will do it later. I do have another problem with this. So that actually gives me the function, the derivative of which is equal to x to the power of a, because x to the power of a plus 1 gives me a plus 1 times x to the power of a, if I differentiate it. And then I have to divide by a plus 1 to get just x to the power of a. Next, e to the power of x. Now, e to the power of x is the only function derivative of which is equal to itself. So I know that this is e to the power of x plus c, obviously. Because the derivative of e to the power of x is equal to e to the power of x. So we know that. Now, these are all very, very trivial things, and they're immediately following from what we know about derivatives. The derivative of a power function and the derivative of exponential function. So there's another exponential function, integral of a to the power of x dx. Now, do you remember what is a derivative of a to the power of x? Well, that's a to the power of x times natural logarithm of a. So I have to neutralize natural logarithm of a to get just a to the power of x. Derivative of this is a to the power of x multiplied by logarithm of a. So I have to divide it by logarithm, natural logarithm of a to get just a to the power of x. Next, let's go to this function. This is exactly x to the power of minus one. Now, one of the few things which I remember about differentiation that logarithm x has a derivative one over x. I do remember power function. I do remember exponential function. I do remember logarithm, and I remember sine and cosine. Everything else I derive from it. So I do remember this logarithm x. If you differentiate it, you will get one over x. Now let's wipe it out and go to the next series. This is a very simple and very short lecture which I have titled simple exercise. So it's very, very simple. It's just sufficient to know some basic formulas of differentiation to come up with whatever I have here. Six, integral of sine of x. Now, what is a derivative? What is the function derivative of which is equal to sine? Well, cosine, right? But cosine gives me minus sine. So I have to get minus cosine x to get minus sine and minus would be plus sine. And that's exactly what it is. Now, similarly, what is function derivative of which is equal to cosine? Again, I do remember it happened to be sine. So I have this. Eight, cosine x minus sine x. Well, obviously you probably guess that it should be some kind of a linear combination between sine and cosine. So if I will take sine to get this and cosine to get minus sine, the derivative of which would be cosine minus sine, right? So that would be an answer. Next two. Okay, integral of hyperbolic sine of x. Remember what it is? I think I did explain it once. This is a to the power of x minus e to the power of minus x divided by 2. And cosine hyperbolic is equal to plus divided by 2. So let's go to derivative. What's the derivative of sine? Well, one half is going down, obviously, right? e to the power of x, derivative is e to the power of x. Now minus, minus derivative of e to the power of minus x. Now that's a chain rule. First we do e to the power of something. This is e to the power of something times derivative of inner function, which is multiplication of minus 1 gives me plus. And which is what? This is cosine. So derivative of hyperbolic sine is a hyperbolic cosine. Now derivative of hyperbolic cosine is e to the power of x. And it would be minus e to the power of minus x and one half. So it would be hyperbolic sine, right? So let's use it. Now we know that this is dx, cosine hyperbolic plus c. And my last example, integral of hyperbolic cosine is hyperbolic sine plus c. Well, it's not an accident that these functions are called sines, hyperbolic but sines. Because they're really behaving very much like real sine and cosine. For instance, real sine has a derivative of cosine and hyperbolic sine has a derivative of cosine. Real cosine has a derivative of minus sine. But, well, minus is just a minor detail. Hyperbolic sine has a derivative of sine. So they're turning into each other by differentiation. And by the way, the formula about sine square plus cosine square equals to one. In this case, it's actually cosine square minus sine square is equal to one. It's very easy to prove. And that was my last example. I do suggest you to go through this again. You can take a look at the Unisor.com website and look at the website. All these examples are there with answers. Try to do them just by yourself. So you remember it a little bit better. Well, other than that, that's it. It's a very short lecture and the real difficulties will come next. Thank you very much. Good luck.