How to prove that $\int\limits_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}=0$?

Question

I am stuck trying to show that $\displaystyle \int\limits_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}=0$

I have tried using a Squeeze Theorem type approach, but at $\pi/4$ any function I choose overlaps and becomes less than or greater than the function depending. I am not sure where to start anymore. Anything will be helpful. Thank you in advance!

NeverBeenHere · Answer 1 · 2013-01-09T15:38:47.297

11

Let $x \mapsto -x$ then we get $\displaystyle I = \int_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^{-x}+1}$ -- adding them we have:

$\displaystyle 2I = \int_{-\pi/2}^{\pi/2} \frac{\cos(2x)}{e^x+1}+\frac{\cos(2x)}{e^{-x}+1}\;{dx}= \int_{-\pi/2}^{\pi/2}\cos(2x) = 0$. Thus $I= 0$.

edited Jan 09 '13 at 15:38

answered Jan 08 '13 at 08:00

NeverBeenHere

933

1

Thank you so much! That was exactly what I was looking for. – Yousuf Soliman Jan 08 '13 at 08:13
@AmrDiab Glad I was of help. :] – NeverBeenHere Jan 08 '13 at 08:18
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This is probably a more fundamental question, but why can you let x be -x for one part only? – Yousuf Soliman Jan 08 '13 at 08:29
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@AmrDiab Not sure what you're asking but $\cos(-2x) = \cos(2x)$ as the cosine function is even and the limits get inverted back to their back to their original positions by the negative from the differential. If you're confused by $x \mapsto -x$ then let $x = -t$ then change $t$ to $x$ (dummy variable) when adding. – NeverBeenHere Jan 08 '13 at 08:35
Ohh okay. I wasn't sure if you could use a dummy variable like that. Thank you again. – Yousuf Soliman Jan 08 '13 at 08:37

score 5 · Accepted Answer · edited Apr 13 '17 at 12:19

5

There is a result stated as follows

Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$

Using this result, $\displaystyle \int_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}dx=\int_{0}^{\pi/2}\cos(2x)\,dx=0$.

edited Apr 13 '17 at 12:19

Community

1

answered Jan 08 '13 at 09:21

pipi

2,461

interesting link. Could you please check if this is really what you intended to link to? – Fabian Jan 08 '13 at 10:10
@pipi We can replace $e^{x}$ with $a^x$ where $a\in\mathbb{R}^{+}$. :] – NeverBeenHere Jan 08 '13 at 12:42
Oh, I've realised that we've already used $a$.
Say then $\alpha^x$ to avoid confusion.
– NeverBeenHere Jan 08 '13 at 12:56
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@Artist We may replace $1+e^x$ by $1\pm \alpha ^{k(x)}$ for any odd function $k(x)$. See here – pipi Jan 09 '13 at 23:42
@pipi Very nice! – NeverBeenHere Jan 10 '13 at 14:43

How to prove that $\int\limits_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}=0$?

2 Answers2