What is gained by learning geometry

Taken from Coxeter:

"..someone who begun to read geometry with Euclid asked him 'What shall I get by learning these things?' Euclid called his slave and said "Give him a dime, since he must make gain out of what he learns."

The Integral of ln (x) and (ln (x))^n

The natural logarithm ln (x) and its inverse function e^(x) have special derivatives which must be memorized because they do not fall under a more general differentiation rule. e^x is easy because its derivative and antiderivative are itself. The derivative of ln (x) is 1/x, but its integral is often left as an exercise in a section about integration by parts. It is generally the first instance of integration by parts where one of the two functions (u and dv or f and g') is the invisible unity, one.

The Integral of ln (x):

IBP Formula

We let u = ln (x) , so du =1/x
And v = x because we let dv = 1dx

The IBP formula becomes:

Thus:

When we take this same approach to the antiderivative of ln^k (x) we begin to see a pattern which yields a simplified integral equation:

First application of integration by parts:

Second application of integration by parts:



And as the pattern continues, we see that:

Guidelines for Finding Domain and Range of a Function (TBC'd)

The domain of a function (say y=f (x)) is the set of x-values that can be inputted into the function which we named 'f', whereas the range is the set of all outputs given by plugging in the valid x-values occurring in the domain.

A simple example could be the function f (x)=x. The domain of this function is anything (all real numbers), and the range will be the same. In calculus (don't care about calculus? Skip to the next paragraph), it's necessary to determine the domain and range of functions for limits, derivatives, and integrals. We devise simple algorithms for computing limits, derivatives, and integrals that may be applied to functions with problematic domains or ranges. After we are we taught these simple processes, we must learn how to deal with hairy functions by identifying the problematic points and adjusting the approach. A good example of this is finding the limit of a function with a hole discontinuity.

Let us illustrate the most common domain issues with examples:

Example 1 (dividing by 0): When we learned about division in grade school we were taught that we can't divide by zero. Why? When we divide a number by another number we are asking how many groups of the latter number make up the first number. 10 divided by 5 is 2, since 2 groups of 5 make 10 (confused? Replace groups of with times and make with equals). If we have some non-zero number divided by zero, infinitely many groups of zero added together will still be zero. Thus, any function that is a quotient will have a problematic domain.

Here's the jist: Any function with a denominator that is zero at an x-value is cannot have that point in its domain.

The domain of f (x)=1/x is all real numbers except x=0.

The domain of f (x)=1/(x^2 - 1) is all real numbers except x=1 and x=-1.

The domain of f (x)=1/(ax^2+bx+1) is all real numbers except x = (-b +/- sqrt (b^2 - 4ac)/ 2a by the quadratic formula solving for ax^2+bx+c=0.

Example 2 (Square rooting negative numbers):

To be continued