Intereting Posts

Power of prime ideal
Direct evaluation of complete elliptic integral
Why the Riemann hypothesis doesn't imply Goldbach?
I still forget concepts even after answering numerous math problems
What is the intersection of all Sylow $p$-subgroup's normalizer?
How to express complex multiplication with a linear operator on $\mathbb R^n$?
Integral $\int_0^{1/2}\arcsin x\cdot\ln^2x\,dx$
Is there a simpler way to find an inverse of a congruence?
$5^n+n$ is never prime?
Counter example for a result of intersection of subspaces
Are $3$ and $11$ the only common prime factors for $\sum_{k=1}^N k!$ for $N\geq 10$?
Compute the characteristic equation 3×3 matrix
What word means “the property of being holomorphic”?
Homeomorphism between the Unit Disc and Complex Plane
draw $\triangle ABC$ in which $AB=5.5$cm, $\angle C =40^{\circ}$ and $BC-AC=2.5$cm

I have a question:

Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that

$$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$

How can I do this? Don’t know how to start.

- Is there a chain rule for integration?
- Is positive the same as non-negative?
- compute the following integral $\int^{a}_{-a} \sqrt{a^2-x^2}dx$
- Are all limits solvable without L'Hôpital Rule or Series Expansion
- minimum and maximum problem
- Convergence of a double sum

- Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $
- Why isn't $f(x) = x\cos\frac{\pi}{x}$ differentiable at $x=0$, and how do we foresee it?
- Calculus conjecture
- Prove that $g(x)=\frac{\ln(S_n (x))}{\ln(S_{n-1}(x))}$ is increasing in $x$, where $S_{n}(x)=\sum_{m=0}^{n}\frac{x^m}{m!}$
- Are there any calculus/complex numbers/etc proofs of the pythagorean theorem?
- What is the inverse of the function $x \mapsto \frac{ax}{\sqrt{a^2 - |x|^2}}$?
- Calculate Inverse Laplace transform
- Need help solving complicated integral $\int e^{-x}\cos4x\cos2x\,\mathrm dx$
- Evaluate $\sum \sqrt{n+1} - \sqrt n$
- how do you do this integral from fourier transform.

You have

\begin{align*}

I &=\int\limits_{-a}^{a}\frac{f(x)}{1+e^{x}} \ dx \qquad\qquad \cdots (1)\\\ I &= \int\limits_{-a}^{a} \frac{f(x)}{1+e^{-x}} \ dx \qquad\qquad \Bigl[ \small\because \int\limits_{a}^{b}f(x) = \int\limits_{a}^{b}f(a+b-x) \ \Bigr] \quad \cdots (2) \\\ \Longrightarrow 2I &= \int\limits_{-a}^{a} \biggl[ \frac{f(x)}{1+e^{x}} + \frac{e^{x}\cdot f(x)}{1+e^{x}} \biggr] \ dx \quad\qquad \cdots (1) + (2)\\\ &=\int\limits_{-a}^{a} f(x) \ dx = 2 \int\limits_{0}^{a} f(x) \ dx \qquad \Bigl[ \small \text{since}\ f \ \text{is even so} \ \int\limits_{-a}^{a} f(x) = 2\int\limits_{0}^{a} f(x) \Bigr]

\end{align*}

$\textbf{Note.}$ A similar problem, which uses result $(2)$ can be found here:

- Integration of a trigonometric function

This works because the even part of $\displaystyle{\frac{1}{1+e^x}}$ is $\frac{1}{2}$.

If $g:[-a,a]\to \mathbb R$ is a function, then $g$ has a unique representation as a sum of an even and an odd function, $g=h+k$, with $h(-x)=h(x)$ and $k(-x)=-k(x)$. If $f:[-a,a]\to\mathbb R$ is even, then $g(x)f(x)=h(x)f(x)+k(x)f(x)$ has even part $h(x)f(x)$ and odd part $k(x)f(x)$. Since the integral of an odd function on $[-a,a]$ is zero and the integral of an even function on $[-a,a]$ is twice the integral on $[0,a]$, this yields

$$\int_{-a}^a g(x)f(x)dx=\int_{-a}^ah(x)f(x)dx=2\int_0^a h(x)f(x)dx.$$

As has been seen in previous questions on this site (like this one) the formula for $h$ is $h(x)=\frac{1}{2}(g(x)+g(-x))$. In this problem, $\displaystyle{g(x)=\frac{1}{1+e^x}}$, and $h(x)=\frac{1}{2}$.

Another way of looking at this, where you are naturally led to the result: Since $f(x)$ is even, what you need to show is that

$$\int_{-a}^a {f(x) \over 1 + e^x}\,dx = {1 \over 2}\int_{-a}^a f(x)\,dx$$

The difference between the left hand side and the right hand side is

$$\int_{-a}^a f(x)\left({1 \over 1 + e^x} – {1 \over 2}\right)\,dx$$

$$= {1 \over 2}\int_{-a}^a f(x) \frac{1 – e^x}{1 + e^x}\,dx$$

Since this is to be zero for *all* even $f(x)$, you’d expect the function $\frac{1 – e^x}{1 + e^x}$ to be odd, so that the product $f(x) \frac{1 – e^x}{1 + e^x}$ would be odd and thus the integral becomes zero. And sure enough, one can verify readily that this function is in fact odd, so that the above integral is always zero.

This was supposed to be a comment to Zarrax’s answer, but it got too long.

Another way to look at Zarrax’s answer goes like this:

We have the expression

$$\frac{1 – e^x}{1 + e^x}=\frac{e^{-\frac{x}{2}} – e^{\frac{x}{2}}}{e^{-\frac{x}{2}} + e^{\frac{x}{2}}}=-\tanh\frac{x}{2}=-\frac{\sinh\frac{x}{2}}{\cosh\frac{x}{2}}$$

Since $\frac{f(x)}{\cosh\frac{x}{2}}$ is even and $\sinh\frac{x}{2}$ is odd, their product is odd. Since $\int_{-a}^a g(x)\mathrm dx=0$ if $g(x)$ is odd, the integral of $f(x)\tanh\frac{x}{2}$ over the interval $[-a,a]$ is zero if $f(x)$ is even.

I’ve been looking at this more closely and it’s now pretty much demystified:

If $k$ is an integer, $f(x)$ is even and $u(x)$ is any uneven function, the integral $\int_{-a}^a$ over $f(x)\,u(x)^{2k+1}$ is trivially zero and so you have

$$\int_{-a}^a \, f(x) \left( \frac{1}{2} + \sum_{k=0}^\infty c_k \, u(x)^{2k+1} \right) = \frac{1}{2}\int_{-a}^a f(x) \,{\mathrm d}x = \int_0^a f(x) \,{\mathrm d}x$$

for any sequence $c_k$.

The family of functions

$$\dfrac{1}{1+e^{u(x)}} = \frac{1}{2} – \frac{1}{4}u(x)+\frac{1}{48} u(x)^3+\dots$$

is one case of a function with such an uneven expansion.

Even more specifically, your case is that faction with $u(x)=x$.

- Find the sum of all the multiples of 3 or 5 below 1000
- Difference between Modification and Indistinguishable
- How many points in the xy-plane do the graphs of $y=x^{12}$ and $y=2^x$ intersect?
- How to calculate the cost of Cholesky decomposition?
- Infinite series $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$
- How to divide a $3$ D-sphere into “equivalent” parts?
- What type of triangle satisfies: $8R^2 = a^2 + b^2 + c^2 $?
- If $f: M\to M$ an isometry, is $f$ bijective?
- Epsilon-n proof confusion
- Showing that $f(x)=x\sin (1/x)$ is not absolutely continuous on $$
- A reference for existence/uniqueness theorem for an ODE with Carathéodory condition
- Calculating percentages for taxes
- Triangle $ABC$ and equilateral triangles $ABC'$, $BCA'$ and $ACB'$.
- Bernoulli Numbers
- Reference for the subgroup structure of $PSL(2,q)$