mathimatikoi.org

Evaluate the integral:

$$\int_{0}^{\pi}\ln^2\left ( a^2-2a\cos x+1 \right )\, {\rm d}x$$

... The time is so past .... One attempt to evaluate the given integral using the sub $x=\tan u$. $$\begin{aligned} \int \frac{dx}{x\sqrt[3]{x^2+1}} &\overset{x=\tan u}{=\! =\! =\! =\!}\int \frac{\sec^2 u}{\tan u\sqrt[3]{\tan^2 u+1}}\, du \\ &\overset{\tan^2 u+1 =\sec^2 u}{=\! =\! =\! =\! =...

Good evening Nickos... Let me answer, for now, integrals \( ii. \,\,\,\, iii. \). \(ii. \) $$\begin{aligned} \int \frac{\sin x}{\sqrt{1+\sin 2x}}\, dx &= \int \frac{\sin x\sqrt{1+\sin 2x}}{\left ( \sin x+\cos x \right )^2}\, dx \\ &\overset{\left ( \ast \right )}{=\!} \int \frac{\tan x\sec^2...

Prove that \[\displaystyle\sum_{n=2}^\infty \frac{(-1)^n\,\zeta(n)}{n\,(n+1)}=\frac{1}{2}(\log2+\log\pi+\gamma-2)\] where \(\zeta\) is Riemann zeta function and \(\gamma\) is Euler–Mascheroni constant. P.S. Mentioned by akotronis on Integral involving $\Gamma$ function . Here is another solution. T...

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an analytic and $1-1$ function and let $\mathbb{D}$ be the open unitary disk. Prove that:

$$\iint \limits_{\mathbb{D}} \left |f'(z) \right| \, {\rm d}z = \text{area}(f \left(\mathbb{D} \right))$$

What eigenvalues is possible for a matrix $P$ to have if $P^2=P^T$ holds.

I am posting this exercise here but perhaps it can fit elsewhere. Show that the integral: $$\begin{eqnarray*}V_n=\int_0^1 \int_0^1 \cdots \int_0^1 \frac{x_1^2+x_2^2 +\cdots +x_n^2}{x_1+x_2+\cdots+x_n}\, {\rm d}x_1 \, {\rm d} x_2 \cdots \,{\rm d}x_n \end{eqnarray*}$$ converges to $2/3$ as $n \righta...

Alan and Barbara play a game in which they take turns filling entries of an initially empty $2008 \times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the re...

Hi Grigoris... here is a second proof. First of all we see that series is convergent, because it is bounded from above from the geometric series that has ratio \( 1/2 \). Let us assume that the sum is rational \( \frac{a}{b} \) and we will be led to a contradiction. Choose \( n \) such that \( b< 2^...

Is the number \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{2^{n^2}} \) rational?