# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

2,352 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
1k views

### Can someone explain this integration trick for log-sine integrals?

I was working on this rather challenging log-sine integral: $$\int_{0}^{2\pi}x^{2}\ln^{2}\left(2\sin\left(x \over 2\right)\right)\,{\rm d}x = {13\pi^{5} \over 45}$$ The upper limit is a waiver ...
536 views

353 views

294 views

### Methodologies to Evaluate $\lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\cos(x^3/3)\,dx$

In This Answer, I wrote "It is straightforward to show that $\displaystyle \lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\,\cos(x^3/3)\,dx=\frac\pi2$." For completeness, I've included the "...
492 views

### Omega Constant Integral

Whilst reading this Math SE post, I saw that the OP mentioned the integral $$\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$$ where $\Omega$ is the unique solution to ...
203 views

183 views

### Calculation of $\int_{0}^1 \frac{\sin(\ln^4(1-x))}{x}~dx$

$$I=\int_0^1 \frac{\sin(\ln^4 (1-x))}{x}dx$$ What is the closed-form evaluation of this integral? I honestly do not have a single clue how to solve this. (There is no application, but it is out of ...
109 views

### How to solve this definite Integral containing $E_{1}${.}!

The integral is: $$\int_{N}^{\infty}\frac{E_{1}(cz+d)}{az+b}e^{-pz}dz$$ where, $E_{1}${.} is the exponential integral, and $$a>0,\ b>0,\ c>0,\ d>0,\ p>0,\ N>0.$$ This is similar ...
205 views

### The minimum of $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx$ is attained for $k=n$

I have the following conjecture: For each given $n\in\mathbb{N},\ n\ge 2$ the minimum of the sequence of integrals $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx,\ k=1,2,\dots,n$ ...
329 views

232 views

### The elementary methods to compute $\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\quad;\quad\text{for}\, \alpha>0$

How to compute the following integral using elementary methods (high school methods). \begin{equation}\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\qquad;\qquad\text{for}\, \alpha>0\end{...
265 views

### Computing the integral $\int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi.$

Integrate $$\int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi.$$ Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ...
### On the closed-form of the triple integral $\int_0^\infty\int_0^\infty\int_0^\infty\frac1{xyz\left(x+y+z+1/x+1/y+1/z\right)^2}\rm{dx\,dy\,dz}$
While doing research for my recent post on the Clausen function $\rm{Cl}_m(x)$, I came across in p. 19 of this paper (by one of the Borwein brothers) the remarkable integral, I_3 =\frac4{3!}\int_0^\...