# Tagged Questions

173 views

### Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$\int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
56 views

### Integral's Closed-form expression in terms of hypergeometric function

I want to solve the following integral: $$I = 2\left[\int_{0}^{1}\dfrac{y^m}{(1 - ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y+\int_{0}^{1}\dfrac{y^m}{(1 + ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y\right]$$ ...
64 views

48 views

77 views

### An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
160 views

### Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
40 views

114 views

### Show that $\int_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}$

I'm looking for a clever way to show that $$\int\limits_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}.$$ All suggestions will be appreciated!
The following problem has appeared in 2013 January qualifying exam in Purdue University, which is publicly available here. Problem 3. Let $\{a_k\}$ be sequence of positive numbers such that ...