Throughout the book the concept of Exponentials and Logs has been developed. In Chapter #1 you reviewed the general characteristics of these functions and in Chapter #4 you learned how to take their derivatives. In this section you will look at how LiveMath can help you understand the integrals of these very important functions.
As you have perhaps already noticed, these functions can be a confusing twisted web of inverses and inverses of inverses. It seems hard to pick a good starting point for their study. There seems to be one huge incestuous relationship between them all, everything seems to be related to every other thing and each function seems to need another for its understanding! Bear in mind that very few readily recall all of the rules and shortcuts that are part of their study. If you are the type that has a hard time remembering math rules, rest assured, you are not alone. This, however, should not hold you back from investigating the mathematics, after all you have LiveMath to help you!
The following section will introduce the more important areas of this subject. In Chapter #1 you learned that the inverse of the Exponential Function, b^x, is the General Log Function log_b(x) and that the very important Exponential Function with the base e (e^x) has an inverse of the Natural Log Function ln(x). You will look at both the General Log and General Exponential functions, but because of their importance in the study of calculus and higher mathematics, you should concentrate your efforts on the Natural Log and Natural Exponential functions. With this in mind, displayed below are the two most important rules for these functions and the ones you should memorize.
The first important relationship is the fact that the Natural log of x, ln(x), has a derivative of 1/x (as you learned in Chapter #4). And conversely, the anti-derivative of 1/x is ln(x).
Secondly, the number e is remarkable in the fact that the derivative of e^x is equal to e^x, and that the integral of e^x conversely is equal to e^x. This relationship, as you have seen before, is one of the most extraordinary properties in all of mathematics!
In Example #4.5 you derived the derivative of the General Exponential Function. It follows that if you take the antiderivative of this new function f'(x) you will get the Exponential Function back (with a constant of integration attached). The following graphic shows this:
The method used to verify this is the following: Since ln(b) is a constant, it can come outside of the integral sign and you are left with solving the integral of b^x.
Using LiveMath to find this you have:
which gives you the answer below (turn Auto Simplify off to see the intermediate step):
The question becomes; How does LiveMath solve the integral of b^x above? To do this, you use the general definition of the derivative of b^x which is re-stated below:
Divide both sides by ln(b) and you get:
Using the fact that you do not disturb the equality by integrating both sides (with respect to x):
Switching the equation around and bringing the constant 1/ln(b) outside of the integral:
And because integration undoes derivatives, simplification gives you:
Of course LiveMath will do this for you, and if you use e for the base, the following develops, giving you the integral of the Natural Exponential Function. Turn off Auto Casing to keep the printout clean, knowing that a constant of integration is part of the following answer:
This may seem like a lot of work to get to what is a stated definition in many books, but by going through the derivation you learn another method you should become familiar with. The method of applying the Integral Op to both sides of an equation. You will use this method many times in your continued studies of calculus and differential equations.
Finding the Integral of the General Log Function becomes an easy job as soon as you know how to integrate the Natural Log function. The secret to finding the Natural Log integral is to use Integration by Parts. As you found in an eariler example, when there seems to be only one function in the integrand you can remember that 1-dx can be the other one used for this method. Using LiveMath derive this Integral.
Now that you know the Integral of the Natural Log Function you can easily determine the General Log integral. To do this all you have to do is use the Change of Base formula introduced in Chapter #4.
Back in Chapter #4 you took a good deal of time finding and defining e and studying the Natural Exponential Function . To integrate these functions, it is necessary that you look at what gives you the Natural Log Function. There are a couple of items you need to clear up concerning this relationship. From the beginning of this section:
Notice that the integral in the equation on the left is a definite integral with the function defined with a different dependent variable (t). This is called a 'dummy variable' and has to with the fact that this definition has its roots in the Fundamental Theorem. Below Livemath solves this Definite Integral. Remember to use the Log Notebook found on your disk.
When looked at closely an interesting question comes to mind. Why wouldn't you use the Power Rule to find the Integral of 1/x. The reason becomes apparent when you solve the integral of x^-1. In the example shown below, you find that the answer delivered is an expression that is undefined (division by 0). If you, however, select the integrand and Simplify and then Integrate, you get the Natural Log function.
The question is; What happens when x < 0. since ln(x) is undefined for x < 0?
This shows us that no matter what x is, the derivative is 1/x. This leads to the fact that when you Integrate you get back the function ln |x| + c.