Associated Topics || Dr. Math Home || Search Dr. Math

### Integral of dx/(1+(tanx)^sqrt2)

```Date: 06/28/2003 at 20:37:26
From: Michael
Subject: A challenging integration

Find the value of the integral that ranges from 0 to pi/2 of dx/(1 +
(tanx)^S), where S is the square root of 2.

I know this question has a solution, but I haven't the slightest idea
how to do it. I tried a substitution letting z = 1 + (tanx)^S, and
that eventually brought me to a rather interesting fraction after
simplifying the trig functions. Nonetheless, no matter what I do, I
never arrive at an answer.

The exponent on the tan(x) is the square root of 2, so it is
intimidating just to look at.
```

```
Date: 06/29/2003 at 09:18:05
From: Doctor Pete
Subject: Re: Extremely challenging integration

Hi Michael,

This question is an old Putnam problem, and the solution is very well
known. The reason that the exponent S = Sqrt[2] was chosen is so that
the integrand is not integrable in closed form. Thus any efforts to
try to substitute or transform the integrand will fail.

The key to solving this question is to use the fact that the integral
is definite, and that the integrand obeys a particular property
which I will describe presently.  If we consider the substitution of
the form

u = Pi/2 - x,
du = - dx,

it is clear that the interval of integration is reversed and the
integrand is negated, the total effect of which is that we again
arrive at a definite integral over [0, Pi/2]. However, the integrand
becomes

du/(1 + Tan[Pi/2 - u]^S),

and since

Tan[Pi/2 - u] = Cot[u] = 1/Tan[u],

we find the integrand is

du/(1 + 1/Tan[u]^S).

Now simplify the fraction to obtain

Tan[u]^S du/(Tan[u]^S + 1).

One should now realize that the value of the integral has not changed,
but the integrand takes on a convenient form. In particular, twice the
integral's value is simply

1/(1 + Tan[u]^S) + Tan[u]^S/(Tan[u]^S + 1) du,

which is just 1 du. Therefore the integral evaluates to Pi/4.

As you can see here, definite integrals can sometimes be evaluated by
somewhat non-traditional means, where an indefinite integral would
fail. This is because the integrand may satisfy certain special
properties over a specific interval of integration. For instance, the
integral of

Log[Sin[x]] dx

on [0, Pi/2] is equal to -Pi Log[2]/2, but there is no elementary
antiderivative of this integrand.

- Doctor Pete, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 06/29/2003 at 13:51:08
From: Michael
Subject: Thank you (A challenging integration)

Thank you so much! The solution is so very easy, but I would never
have thought of that in a million years. This is one of those times
when you realize how beautiful math really is. Thank you so much, the
answer makes perfect sense to me now!
```
Associated Topics:
College Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search