# Tagged Questions

Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

50 views

### Convergence of the integral $\int_0^\infty \frac{\ln^3(1+x^{1/4})}{x^{1/5}+x^2}\,\arctan(x)\;dx$

How to prove that this integral is convergent? $$\int_0^\infty \frac{\ln^3(1+x^{1/4})}{x^{1/5}+x^2}\,\arctan(x)\;dx$$ I have a little experience with this kind of problem, I know we should solve the ...
61 views

### What to know about convergence of integrals

According to the values of p>0 examine the convergence of the integral: $$\int_0^{+\infty} \dfrac{\ln(1+2x^{3p})}{(x+x^2)^{4p}\arctan(x)^{1/2}}dx$$ I didn't find a good explanation about this kind of ...
75 views

### integration of $\int_{1}^{\infty } \,\left(\frac{2x^{2}+bx\text{+}a}{x(2x+a)} -1\right) \, dx=1$

i need help for this problem; Find values of a and b $$\int_{1}^{\infty} \left( \frac{2x^{2}+bx+a}{x(2x+a)} -1\right) \, dx=1$$ I very appreciate your comments and suggestions.
35 views

54 views

### Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$

I know that the improper integral $$\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$$ is convergent for $s>0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can ...
118 views

### Numerical integration with matrices

I have a matrix integration problem. It is based on the first integral under the section, "energy transfer efficiency and transport time" in the article, environment-assisted transport. There is a ...
84 views

41 views

### Limit of an integral function

I'm stuck with this exercise: Let $f\colon (0,+\infty) \to \mathbb{R}$ such that $f(x)=\int_{x}^{x+\sin{x}}\frac{dt}{\log(1+t)}.$ Prove that $\lim _{x \to +\infty} f(x)=0.$ All I have found is ...
51 views

42 views

I'm trying to calculate the integral $\int_0^1 \frac{dx}{\sqrt{-\ln(x)}}$ using Euler integrals ($\Gamma(x)$ and $B$(x,y)$). I basically have to find a way to make that integral resemble one of the ... 1answer 30 views ### Integral equality involving partial derivatives Update 4: I found the following, updated integral identity: $$\int_{l=-\infty}^\infty l \left. \left( \frac{\partial}{\partial x} f(x,y)\right) \right\vert_{x=y=l} \mathrm{d} l = -\int_{l=-\infty}^\... 3answers 110 views ### Find whether \int_{1}^{\infty} \frac{sin(x)}{x} dx is converging or not Find whether \int_{1}^{\infty} \frac{sin(x)}{x} dx is converging or not. I tried to use comparison test or limit comparison test but could't find a suitable function. How can I determine what type ... 1answer 102 views ### For which values of p,q does the integral \int_0^1 x^p (\ln\frac{1}{x})^qdx converges? For which values of p,q does the integral \int_0^1 x^p (\ln\frac{1}{x})^qdx converges? I use the substitution t=1/x to obtain this better looking integral: \int_1^\infty \frac{(\ln t)^q}{t^{p+... 2answers 257 views ### Easier ways to prove \int_0^1 \frac{\log^2 x-2}{x^x}dx<0 Prove that$$\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$$One way to do this is use the idea in the proof of Sophomore's dream. We have$$x^{-x}=\exp(-x\log x)=\sum_{n=0}^\infty\frac{(-1)^nx^n\log^n x}{... 1answer 38 views ### Using$g(x)=1$for the quotient test of convergence of improper integrals Is it ok to set$g(x)=1$for the quotient test of convergence of improper integrals? I find it easily solves many problems, for example, show if the following converge or diverge:$\displaystyle\...
$$I(a)=\displaystyle\int_{0}^{1} \frac{tan^{-1}ax}{x\sqrt{1-x^{2}}} dx$$ By using Leibniz's formula, I'(a)=\displaystyle\int_{0}^{1} \frac{\partial}{\partial a} \frac{tan^{-1}ax}{x\sqrt{1-x^{2}}} ...