Evaluate Integral of x^2/(1 + e^sin(x))

[Gregg]

Homework Statement

Evaluate $$\int^2_{-2} {x^2dx\over {1+e^\sin x}}$$

The Attempt at a Solution

No clue, tried a series expansion but it shed no light.

 

[Gregg]

This seems to be true

$$\int_{-a}^a{x^2dx \over {1+e^{\sin x} } } = {a^3\over 3}$$

Not sure how to get there though

 

[Gregg]

$$\int_{-2}^{2}\frac{x^2}{1+e^{\sin x}} dx$$

is the same as

$$\int_{0}^{2} \frac{x^2}{1+e^{\sin x}} dx+ \int_{0}^{2}\frac{x^2}{1+e^{\sin -x}}dx$$

Series expansion shows that

$$\frac{x^2}{1+e^{\sin x}} = \frac{x^2}{2}-\frac{x^3}{4}+\frac{x^5}{16}-\frac{7 x^7}{480} \cdots$$

$$\int \frac{x^2}{1+e^{\sin x}} dx= \frac{x^3}{6}-\frac{x^4}{16}+\frac{x^6}{96}-\frac{7 x^8}{3840}+\cdots$$

$$\frac{x^2}{1+e^{\sin -x}} = \frac{x^2}{2}+\frac{x^3}{4}-\frac{x^5}{16}+\frac{7 x^7}{480} \cdots$$

$$\int \frac{x^2}{1+e^{\sin -x}}dx = \frac{x^3}{6}+\frac{x^4}{16}-\frac{x^6}{96}+\frac{7 x^8}{3840}-\cdots$$

$$\int \frac{x^2}{1+e^{\sin x}} dx+ \int \frac{x^2}{1+e^{\sin -x}}dx = \frac{x^3}{3}$$

That's as close as I can get. Before integration, changing x for -x preserves only the first term with an even index. Not really sure how to show that. I doubt this is the way the question is supposed to be answered.

 

[Count Iblis]

1/[1+exp(y)] + 1/[1+exp(-y)] =

1/[1+exp(y)] + exp(y)/[exp(y)+1] =

[1+exp(y)]/[1+exp(y)] = 1

 

[Gregg]

hmmm so

x^2/(1+e^sinx) + x^2/(1+e^sin-x) = x^2. No need for expansion... I think it's a bit sneaky to expect you to notice the symmetry/assymetry in the integral etc.

 

[Count Iblis]

hmmm so

x^2/(1+e^sinx) + x^2/(1+e^sin-x) = x^2. No need for expansion... I think it's a bit sneaky to expect you to notice the symmetry/assymetry in the integral etc.

Most people who know this would have noted it via the well known fact that the Bernouilli numbers B_n for odd n are all zero except for n = 1:

http://en.wikipedia.org/wiki/Bernoulli_number

This fact follows most easily by using the generating function of the Bernoulli numbers:

$$\frac{x}{\exp(x) - 1}=\sum_{n=0}^{\infty}\frac{B_{n}}{n!}x^n$$

and then using the same symmetry.

 

[Count Iblis]

Or via the similar symmetry of the Fermi distribution function around the Fermi energy. That function has a plus 1 in the numerator, so its more similar to this problem.