An integral with log. and arctan

Prove that:

$$\int_{0}^{1}\ln (1+x^2) \arctan x \, {\rm d}x = \ln 2 - \frac{\pi}{2} - \frac{1}{4}\ln^2 2 + \frac{\pi^2}{16} + \frac{\pi \ln 2}{4}$$

Solution

A sinh integral

Let $s>1$. Evaluate the integral:

$$\mathcal{J}=\int_0^\infty \frac{x^{s-1}}{\sinh x} \, {\rm d}x$$

Solution

Integral with fractional part

Let $n \in \mathbb{N}$ and let $\{ \cdot \}$ denote the fractional part. Prove that:

$$\int_0^1 x^n \left\{ \frac{1}{x} \right\}^n \, {\rm d}x = 1- \frac{1}{n+1} \sum_{k=2}^{n+1} \zeta(k)$$

where $\zeta$ is the Riemann zeta function.

Solution

A generalized integral

Let $m, n \in \mathbb{N}_{n \geq 2}$. Prove that:

$$\int_{0}^{1}\frac{1-x^{n-1}}{\left ( 1-x \right )\left ( 1-x^n \right )}\left ( -\ln x \right )^{m-1}\, {\rm d}x= \left ( 1-\frac{1}{n^m} \right )\Gamma(m)\zeta(m)$$

Solution

Trigonometric mixed with exponential integral

Let $a>0$. Prove that:

$$\int_{0}^{\infty}e^{-2ax} \left(\frac{\sin(x)}{x}\right)^{2}\, {\rm d}x=a\log\left(\frac{a}{\sqrt{1+a^{2}}}\right)+\arccot a$$

Solution

A relation with complex unity and logs.

Let $n \in \mathbb{N}$ then prove that:

$$\int_0^{\pi/2} (\log \cos x+ \log 2 + ix)^n \, {\rm d}x + \int_0^{\pi/2} (\log \cos x + \log 2 - ix)^n \, {\rm d}x=0$$

Solution

Definite integral with parameter

For the values of $a$ for which the integral converges, evaluate the integral:

$$\mathcal{J}=\int_0^1 \frac{1+x^a}{(1+x)^{a+2}}\, {\rm d}x$$

Solution

An $n$ dimensional integral

Let $n \geq 3$ be a natural number. Evaluate the integral:

$$\int \limits_{[0, 1]^n} \left \lfloor x_1+x_2+\cdots+x_n \right \rfloor\, {\rm d}\left ( x_1, x_2, \dots, x_n \right )$$

Log. integral

Evaluate the integral:

$$\int_{0}^{1}\frac{x\ln x}{\sqrt{1-x^2}}\, {\rm d}x$$

A squared sum and integral

Prove that:

$${\large \int_{-\pi}^{\pi}}\left ( \sum_{k=1}^{2014}\sin (kx) \right )^2\,dx=2014\pi$$

Solution

An inverse tanh integral

Evaluate the integral:

$$\int_{0}^{1}\sqrt{4x-4x^2} \cdot \operatorname{arctanh} \left ( \sqrt{4x-4x^2} \right )\, {\rm d}x$$

Solution

Integral with trigonometric

Let $a>0$. Evaluate the integral:

$$\int_0^\infty \frac{\sin^2 x}{x^2 \left(x^2+a^2 \right)}\, {\rm d}x$$

Solution

Improper integral with sinus

Prove that:

$$\int_0^\infty \frac{\sin^2 x}{x^2}\, {\rm d}x = \frac{\pi}{2}$$

Solution

Fresnel integrals

Prove that:

$$\int_0^\infty \cos x^2 \, {\rm d}x = \int_0^\infty \sin x^2 \, {\rm d}x =\frac{1}{2}\sqrt{\frac{\pi}{2}}$$

Solution

Definite integral

Prove that:

$$I_{n}=\displaystyle\int_{0}^{2\alpha}{x^{n}\sqrt{2\alpha{x}-x^2}\;dx}=\frac{\pi\,\alpha^{n+2}}{2}\mathop{\prod}\limits_{k=2}^{n}{\frac{2k+1}{k+2}}\,,\quad n\in\mathbb{N}\,,\;\alpha>0$$

Solution 

Integral with digamma

Let $\psi(x)$ denote the digamma function. Evaluate the integral:

$$\int_0^1 \psi(x) \sin 2n \pi x \, {\rm d}x \quad , \qquad n \in \mathbb{N} $$

Solution

Improper integral with logarithm

Evaluate the integral:

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

Solution

Logarithmic integral

Evaluate the integral:

$$\int_{0}^{1}\frac{\ln (1-x) \ln x}{1-x}\, {\rm d}x$$

Solution

Integral and trilogarithm

Evaluate the integral:

$$\int_{0}^{1}\frac{\ln^2 x}{2-x}\, {\rm d}x$$

Solution

A generalized form of an integral

Continuing the post from here we give the generalization which is due to Ovidiu Furdui. 

Let $k$ be a positive integer. Prove that:

$$\int_{0}^{1}\ln^k (1-x)\ln x\, {\rm d}x =(-1)^{k+1}k! \left ( k+1 - \zeta(2)-\zeta(3)- \cdots -\zeta(k+1) \right )$$

where $\zeta$ is the Riemann zeta function.

Solution