Poisson integral

Evaluate the integral:

$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$

Solution

A very interesting integral

Evaluate the integral:

$$\mathcal{J} =\int_0^1 \left( \frac{1}{1-x} + \frac{1}{\ln x} \right) \, {\rm d}x$$

Solution

Integral with log and trigonometric

Prove that

$$\int_0^{\infty} \frac{\log t (1-\cos t)}{t^2}\, {\rm d}t = \frac{\pi}{2} (1-\gamma)$$

where $\gamma$ stands for the Euler - Mascheroni constant.

Solution

Integral with logarithm

Evaluate the integral:

$$\mathcal{J}=\int_0^1 \frac{1-x}{(x+1) \log x}\, {\rm d}x$$

Solution

$\int_{0}^{\pi/2} \frac{\log^2 \left ( \tan x \right )}{\sin^2 \left ( x- \frac{\pi}{4} \right )} \;dx$

Evaluate the following integral:

$$\int_{0}^{\pi/2} \frac{\log^2 \left ( \tan x \right )}{\sin^2 \left ( x- \frac{\pi}{4} \right )}  \, {\rm d}x$$

Solution

Integral with dilogarithm

Let ${\rm Li}_2$ denote the dilogarithm function. Evaluate the integral

$$\int_0^\infty \frac{{\rm Li}_2(-x)}{1+x^2}\, {\rm d}x$$

Solution

Example of a function

Give an example of a continuous function $f:[1, +\infty) \rightarrow \mathbb{R}$ such that $f(x)>0$ forall $x \in [1, +\infty)$ and $\bigintsss_{1}^{\infty} f(x) \, {\rm d}x$ converges while $\bigintsss_{1}^{\infty} f^2(x) \, {\rm d}x$ diverges.

Solution

Double exponential integral

Evaluate the integral:

$$\int_0^{\infty} \int_0^{\infty} \frac{{\rm d}x \; {\rm d}y}{\left(e^x+e^y\right)^2}$$

(Ovidiu Furdui)

Solution

An arctan integral

Evaluate the integral:

$$\mathcal{J}=\int_0^1 \frac{\arctan x}{x \sqrt{1-x^2}}\, {\rm d}x$$

Solution

A weird infinite product

Evaluate the following product

$$P =\left({\frac{2}{1}}\right)^{1/8}\cdot\left({\frac{{2\cdot 2}}{{1\cdot 3}}}\right)^{3/16}\cdot\left({\frac{{2\cdot 2\cdot 2\cdot 4}}{{1\cdot 3\cdot 3\cdot 3}}}\right)^{6/32}\cdot\left({\frac{{2\cdot 2\cdot 2\cdot 2\cdot 4\cdot 4\cdot 4\cdot 4}}{{1\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 5}}}\right)^{10/64}\cdots$$

Solution

$\int_0^\infty \frac{\sin^2 (\tan x)}{x^2}\, dx$

Evaluate the integral:

$$\int_0^\infty \frac{\sin^2 (\tan x)}{x^2}\, {\rm d}x$$

Solution

Parametric integral with logarithm

Let $n \in \mathbb{N} \setminus \{1\}$. Prove that:

$$\int_{0}^{\infty} \frac{n^2 x^n \log x}{1+x^{2n}} \, {\rm d}x = \frac{\pi^2}{4} \frac{\sin \left ( \frac{\pi}{2n} \right )}{\cos^2 \left ( \frac{\pi}{2n} \right )}$$

Solution

$\int_0^{\pi/2} \frac{\log \cos x}{\tan x }dx$

Evaluate the integral:

$$\int_{0}^{\pi/2} \frac{\log \cos x}{\tan x}\, {\rm d}x$$

Integral with logarithm and exponential

Let $k$ be a positive integral. Evaluate the integral:

$$\int_{0}^{\infty}\frac{e^x-1}{e^x+1}\log^k\left(\frac{e^x+1}{e^x-1}\right)\,{\rm d}x$$

(Ovidiu Furdui)

Solution

Logarithmic improper integral

Let $n \in \mathbb{N}$. Find a closed form for the integral:

$$\mathcal{J}=\int_0^\infty \frac{\log^n x}{1+x^2}\, {\rm d}x$$

Asymptotic behavior of Wallis integrals

Let us denote with $W_n$ the Wallis integral, that is:

$$W_n=\int_0^{\pi/2} \sin^n x \, {\rm d}x$$

Prove that $W_n \sim \sqrt{\frac{\pi}{2n}}$.

Solution

A definite symmetric integral

Evaluate the integral:

$$\int_{-1/\sqrt{3}}^{1/\sqrt{3}}\left(\frac{x^4}{1-x^4}\right) \arccos \left(\frac{2x}{1+x^2}\right)\, {\rm d}x$$

Solution

A definite integral

Evaluate the integral:

$$\mathcal{J}= \int_{-\pi/2}^{\pi/2} \frac{x^2\cos x}{1+x+\sqrt{x^2+1}}\, {\rm d}x$$

Solution

A squared log. integral

Prove that:

$$\int_{0}^{\infty}\frac{(x+1)\ln^{2}(x+1)}{(4x^{2}+8x+5)^{3/2}}dx=\frac{\pi^{2}}{40}$$

Solution

A closed form for a generalized sin integral

Let $n \in \mathbb{N}$. Deduce a closed form for the integral:

$$\mathcal{J}_n=\int_0^\infty \left( \frac{ \sin x}{x} \right)^n \, {\rm d}x$$

Solution