An integral involving Sophomore's dream constant

The Sophomore's dream constant denoted as $\mathscr{S}$ is defined to be the series:

$$\mathscr{S}= \sum_{n=1}^{\infty} \frac{1}{n^n}$$

One interesting identity is the following

$$\int_0^1 x^{-x}\, {\rm d}x = \sum_{n=1}^{\infty} n^{-n} = \mathscr{S} \tag{1}$$

In this thread however we take the above for granted (and leave the proof to the reader since it is not that difficult)  and prove another interesting identity which is:

$$\int_0^1 \frac{\ln x}{x^x}\, {\rm d}x =- \mathscr{S}$$

Solution

An integral with arctan

Evaluate the following integral:
$$\int_0^{2-\sqrt{3}}\frac{\arctan x}{x}\, {\rm d}x$$

The result is due to Ramanujan.

Solution

Improper integral with logarithm

Evaluate the integral:

$$\int_0^1 \frac{\ln x}{1+x^2}\, {\rm d}x$$

Solution

Integral with arctangents

Evaluate the integral:

$$\int_0^\infty \frac{\tan^{-1}(\pi x)-\tan^{-1} x}{x}\, {\rm d}x$$

Solution

Evaluation of Integral

Evaluate the integral:

$$\int_0^{\infty} \frac{x^3}{e^x-1}\, {\rm d}x$$

Solution

On a classic integral

Evaluate the integral:

$$I=\int_{-\pi/2}^{\pi/2}\frac{1}{2007^x +1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x +\cos^{2008}x}\, {\rm d}x$$

Solution

Definite integral

Let $a>1$. Evaluate the integral:

$$\int_1^{a^2} \frac{\ln x}{\sqrt{x}(x+a)}\, {\rm d}x$$

Solution

Improper integral

Evaluate the integral:
$$\int_0^1 \ln x \ln (1-x) \, {\rm d}x$$

Solution

An integral with arccosine

Evaluate the integral:

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

Solution

Logarithmic integral and Euler sums

Evaluate:
$$\int_0^1 \frac{\ln (1+x^2)}{1+x}\, {\rm d}x$$

Solution

Integral with floor function

Evaluate the integral:

$$I=\int_{0}^{1}\left ( \left \lfloor \frac{2}{x} \right \rfloor-2\left \lfloor \frac{1}{x} \right \rfloor \right )\, {\rm d}x$$

Solution

Integral with logarithm

Let $n \in \mathbb{N}, \; n >2$  then prove that:

$$\int_{0}^{\infty}\frac{\log \left(\frac{1}{x}\right)}{\left ( 1+x \right )^n}\, {\rm d}x = \frac{1}{n-1}\sum_{k=1}^{n-2}\frac{1}{k}$$

Solution

Coxeter integral

Evaluate:

$$\int_{0}^{\pi/4}\tan^{-1}\sqrt{\frac{\cos 2\theta}{2\cos^2 \theta}}\, {\rm d}\theta$$

Solution

Integral arising from Fourier series

Evaluate:

$$\int_0^{2\pi} \ln^2 \left( 2 \sin \frac{x}{2} \right)\, {\rm d}x$$

Solution

Improper integral involving floor function

Evaluate the integral:

$$\mathcal{L}_1=\int_{0}^{\infty}\left \lfloor x \right \rfloor e^{-x}\, {\rm d}x$$

Solution

Integral of floor function

Evaluate the integral:

$$K_n = \int_{-n}^{n} \left \lfloor x \right \rfloor\, {\rm d}x$$

Solution

Curve and line integrals

Let $\gamma$ be defined as $\gamma(t) = e^{-t} (\cos t, \sin t ), \;\; t \geq 0$.

a) Sketch the graph of the curve.
b) Evaluate the line integrals:

$$ \begin{matrix} &  \displaystyle ({\rm i})\; \oint_{\gamma}\left ( x^2 +y^2 \right )\, {\rm d}s &  & ({\rm ii}) \displaystyle \oint_{\gamma} (-y, x)\cdot {\rm d}(x, y) \end{matrix}$$

Solution

From Russian Mathematical Olympiad

Evaluate:

$$  \int_0^1 \frac{\arctan \dfrac{x}{x+1}}{\arctan \dfrac{1+2x-2x^2}{2}}\,\,{\rm d}x$$

Solution:

Improper integral

Let $n \in \mathbb{N}$. Evaluate the integral:

$$\int_0^\infty \frac{{\rm d}x}{(1+x^2)^n}$$

Solution:

Integral involving a logarithm

Evaluate the integral:
$$\int_{0}^{\infty}\frac{1}{x}\ln \left ( \frac{x^2+2kx \cos b+k^2}{x^2+2kx \cos a+k^2} \right )\,dx $$

whereas $0\leq a, b \leq \pi, \;\; k>0$.

Solution: