Name: Kaushal Shah
Who is asking: Student
Level: Secondary

Question: How Do WE Integrate the following Functions,

0. Integral xtanx dx
1. How was natural base "e" discovered and why e=2.7.......

If you try integrating by parts, the natural choice is to put u = x, du = dx, dv = tan(x)dx, v = ln|cos(x)|.

You end up having to integrate ln|cos(x)|dx, which is not obvious. Something is wrong here: integration by parts is supposed to reduce the integral to something simpler, but here it becomes more complex.

You may try other methods, or with experience, you may come to suspect that there is no closed form for this integral. In the good old days, you would verify the latter by checking up integration tables, but nowadays most people rely on computer algebra packages like mathematica. Here is the outcome:

In[2]:= Integrate[x*Tan[x],x]

        I  2              2 I x    I              2 I x
Out[2]= - x  - x Log[1 + E     ] + - PolyLog[2, -E     ]
        2                          2

Here, I is the square root of -1, E is the base of natural logarithms, Log is the usual ln function and PolyLog(n,z) is the summation from k = 0 to infinity of (z^k)/(kⁿ). (That is, the "polylogarithm function".)

This is beyond the standard calculus courses. The integral of x tan(x) cannot be expressed in terms of polynomials, rational functions, logs, exponentials or trig functions. If you had it as a question in an introductory class it might have been a typo, or it might come from a double integral and you should have chosen the other order of integration.

How was natural base "e" discovered & why e=2.7.......

John Napier discovered logarithms. There is a good account of what he did at the site

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html

The following quotation explains a bit how "e" comes about in there:

"Unlike the logarithms used today, Napier's logarithms are not really to any base although in our present terminology it is not unreasonable (but perhaps a little misleading) to say that they are to base 1/e. Certainly they involve a constant 10⁷ which arose from the construction in a way that we will now explain. Napier did not think of logarithms in an algebraic way, in fact algebra was not well enough developed in Napier's time to make this a realistic approach. Rather he thought by dynamical analogy. Consider two lines AB of fixed length and A'X of infinite length. Points C and C' begin moving simultaneously to the right, starting at A and A' respectively with the same initial velocity; C' moves with uniform velocity and C with a velocity which is equal to the distance CB. Napier defined A'C' (= y) as the logarithm of BC (= x), that is

y = Nap.log x.

Napier chose the length AB to be 10⁷, based on the fact that the best tables of sines available to him were given to seven decimal places and he thought of the argument x as being of the form 10².sin X."

In particular, this makes it clear that it is difficult to talk of ancient mathematics in modern terms, because the difference in notation affects our perceptions a lot.