Math-IA-3.docx - Interesting definite and improper...

Interesting definite and improper integrals
INTRODUCTION
Most of the integrals that are analytical have proved to have interesting applications in the outside
world. Yet, there are integrals which have, important applications in the real world, but are counter-
intuitive. These integrals often do not have an analytical solution in terms of standard functions, but
have a finite value for the improper integral. For instance the integrals,
∫
0
∞
( lnx )
2
(
x
2
+ 1
)
2
dx
and
∫
0
∞
lnx
( x
2
+ 1 )
dx
, are not analytical in terms of standard functions, yet, astonishingly, a finite value is
obtained for the improper integrals ranging from 0 to infinity. In contrast, an analytical solution to the
integral
∫
tan
n
x dx
, where n is a real number, could be found by employing special methods.
However, it is not at all obvious how such an integral could produce an analytical solution. These
observations left me baffled and it came to thought that such integrals would prove interesting to
investigate.
In this exploration I would, in the first section, discuss the integral of
∫
tan
n
x dx
, specifically when
n is a rational number, and attempt to find a value for the improper integral from 0 to infinity. In the
second section I would prove that an analytical solution for
e
x
2
and
cosx
x
cannot be arrived at,
in terms of standard functions, but I will attempt to provide an alternate solution using series

expansions. Consequently, I would explore different methods to find the value for the improper
integrals.
S
ECTION
1
Problem 1 let’s consider the function
f
(
x
)
= tan
n
x
, where n
∈
ℝ .
For n=1, the standard tangent function could be integrated to give
– ln
|
cosx
|
.
This could be
simplified to give
ln
|
secx
|
. However, when n approaches rational numbers; the integral turns
rather complex.
Let us then let n be
1
2
. The integral could be expressed as
∫
√
tanx dx.
The substitution for such
an integral is very hard to deduce. In order to evaluate this integral we will define two supplementary
functions,
I
1
and
I
2
, such that the sum or the difference would result in the integral we are
interested in.

Let:
I
1
be
√
tanx
( ¿ +
√
cotx ) dx
∫
¿
I
2
be
(
√
tanx −