 Hello, everyone. Welcome to this introduction to integration and anti-differentiation. Recall that up until now you have been given a function and asked to find its derivative. Now you are going to be given the derivative and asked to find the original function. This process is called anti-differentiation. Differentiation and anti-differentiation are inverse operations of each other, just as addition and subtraction are, multiplication and division are, and squaring a quantity and taking the square root of that quantity are inverse operations. Anti-differentiation is the process of finding the set of all anti-derivatives of a given function. You are going to notice throughout the problems that you are always asked to find an anti-derivative, not the anti-derivative. There's a reason for that. In general, and we read this that you see on the left side of the equal sign, as the anti-derivative of f of x dx is equal to capital F of x plus c. Let's dissect the different parts of this. F of x is what we refer to as the integrand. Think of another term you have learned in mathematics, the radicant, which is the number underneath a radical symbol, a square root symbol for instance. The number you are taking the root of is what we always called the radicant. Here the function of which you are taking the anti-derivative is called the integrand. The dx that you see there is what we refer to as the differential. It indicates the variable of integration. Capital F of x is the anti-derivative. Finally, the plus c is our constant of integration. Remember how we said before that differentiation and anti-differentiation are inverse operations of each other. Therefore, it will be true that the derivative of the anti-derivative of f of x dx simply is f of x. In accordance with the fact we know differentiation and anti-differentiation are inverse operations of each other. The derivative and the anti-derivative essentially undo each other. The power rule for integration therefore states that if n is an element of the rational numbers, remember q is the mathematical symbol for the rational numbers because q is for quotient and a rational number is a number that can be expressed as a fraction, as a quotient. So if n is an element of the rational numbers but n cannot equal negative one, hopefully it makes sense as to why, we'll talk a little bit more about that in a little bit, then the anti-derivative of x to the n dx is equal to x raised to the n plus one power divided by n plus one. So you're adding one onto the exponent and then dividing by that new exponent plus c. This is the rule that hopefully you figured out for yourself through our introductory activity. A couple things to note though. Consider the anti-derivative of x to the negative one dx which can be rewritten as the anti-derivative of one over x dx. You cannot use the power rule for integration on this because if you add one to the exponent and then divide by that new exponent, you end up dividing by zero which we all know you cannot do. That means we need to come up with another way to get the rule for the anti-derivative of this function. Thinking of the fact that derivatives and anti-derivatives are inverses of each other, can you think of a function that you take the derivative of and get one over x as the answer? Hopefully you thought and remembered that if you take the derivative of natural log of x, you get one over x. Therefore, the anti-derivative of one over x dx is the natural log. Now we do say it's the absolute value of x. There is a reason for that which we will investigate in one of the later lessons, plus c. This right here becomes the first rule you need to memorize, one of the special anti-derivative rules. The anti-derivative of one over x dx is natural log of absolute value of x plus c. Let's consider exponential functions with the number e. Remember that the derivative of e to the x is e to the x. We talked about that it's the only function which is its own derivative. Now let's think about anti-derivatives though. Can you think about what the anti-derivative then of e to the x dx would be? In other words, of what function is e to the x dx, the derivative, or what do you take the derivative of and get e to the x as the answer? Hopefully you figured this one out. The anti-derivative of e to the x dx is simply e to the x plus c. This is the second rule you need to memorize. I like to think it's a pretty easy one. e to the x is a very very special function. Again it's the only function which is its own derivative. It's also its own anti-derivative. What about exponential functions in general though? What if we want to discover a rule for the anti-derivative of e to the x dx? e would be a number. So think what function do you take the derivative of and get e to the x dx as the answer? This one's a little trickier. You have to think a little bit about this and be a little bit more creative. We find that the anti-derivative of e to the x dx is 1 over natural log of a times e to the x plus c. This is the third rule you need to memorize. Let's look at a little proof as to why this is. Remember that derivatives and anti-derivatives are inverses. So if we were to take the derivative of this answer, of the anti-derivative, we should get that integrand back again. That's one of the great things about anti-derivatives that we didn't have with derivatives. A way to actually check our answers. So let's take the derivative of this and see what we get. 1 over natural log of a, that's really a number. It's a numerical quantity so that's our coefficient that would just remain as we take our derivative. Now we need the derivative of a to the x. Think back to your derivative rules. The rule for derivative of a to the x is a to the x times the natural log of a. But look what happens. The natural log of a's cancel out leaving you with a to the x, which is our integrand. As you work through anti-derivative problems on your own, oftentimes you will have to rewrite that which is to be anti-differentiated by using some algebraic manipulation. For example, if we wanted to do the anti-derivative of 5 over square root of x dx, the 5, just like a constant when you're doing derivatives, can simply come out in the front. We're going to bring that square root of x up from the denominator and rewrite it as x to the negative one half. Then we can apply the power rule for integration. Similarly, if we wanted the anti-derivative of cube root of x times the quantity x minus 6 dx, we're going to think of the cube root of x as x to the one third and distribute it over that quantity. And we arrive at x to the four thirds minus 6x to the one third. Once again, you then apply the power rule for integration. Finally, one last example. If we wanted the anti-derivative of x to the seventh plus four over x square dx, we're going to apply the laws of exponents and rewrite this as x to the fifth plus four x to the negative second. Very similar to how you would have worked with it if you wanted to take a derivative.