Articles of integration

Euler-Mascheroni constant: understanding why $\lim_{m\rightarrow \infty} \sum_{n=1}^{m} (\ln (1 + \frac{1}{n})-\frac{1}{n+1})= 1 – \gamma$

I am trying to understand why the Euler-Mascheroni constant $\displaystyle \gamma = \lim_{n \rightarrow \infty} \left ( \sum_{k=1}^n \frac{1}{k} – \ln n \right )$ is equal to $1 – \displaystyle \int_{1}^{\infty} \frac{t-[t]}{t^2}\; \mathrm{d}t$ where $[t]$ is the floor function. There has been an answered question here already: Integral form for the euler-mascheroni gamma constant using […]

Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of the Riemann integral. I want some hint or example, Im unable to find a sequence like this. My work at this moment: If the integral converges to […]

Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.

Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$. The contour is $\gamma=\gamma_1+\gamma_2+\gamma_3$ where $\gamma_1(t)=t$ for $0\leq t \leq R$, $\gamma_2(t)=Re^{i\frac{2\pi}{3}t}$ for $0\leq t \leq 1$, and $\gamma_3(t)=(1-t)Re^{i\frac{2\pi}{3}}$ for $0\leq t \leq 1$. So, the contour is a wedge, and by letting $R\rightarrow \infty$ we’re integrating over one […]

It is an easy question about integral,but I need your help.

How to compute this integral? $$ \int^{\pi}_{0} \frac{\sin(nx)\cos\left ( \frac{x}{2} \right )}{\sin \left ( \frac{x}{2} \right ) } \, dx$$ I need your help.

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\frac1n\sum\limits_{i=1}^n f(a_i) \to \lambda(f):=\int_0^1 f(x)\,dx$ for all continuous $f:[0,1]\to \mathbb{R}_{\ge 0}$. In the proof I used the fact that if $T_g(x)=(g+x) \pmod 1$, then $\lambda(f\circ T_g)=\lambda(f)$ for all continuous […]

Why $\int_0^1(1-x^4)^{2016}dx=\prod_{j=1}^{2016}\left(1-\frac1{4j}\right)$?

In this answer it states that: $\int_0^1(1-x^4)^{2016}dx=\prod_{j=1}^{2016}\left(1-\frac1{4j}\right)$ How to prove that?

Motivation behind Euler substitution in Integrals

As the title suggests, I want to know the motivation behind the Euler substitutions.How someone arrived at these substitutions. To remove any doubts, by euler substitutions i mean – If we have a rational function of $x$ and $\sqrt{ax^2+bx+c}$ , Case I. If $a>0$ , put $\sqrt{ax^2+bx+c}=t \pm x\sqrt{a}$ Case II. If $c>0$ , put […]

How do you find the formula for an area of the circle through integration?

I got this question on my Maths exam today, and the Department of Education has stated it is on the syllabus, but none of the three textbooks I could get my hands mention anything about it. One mentions the volume of a cone, but none the area of a circle. Use integration methods to establish […]

Finding the Moment Generating Function for a random variable X^2

If X~(0,1), integrate to find the moment generating function of a random variable $X^2$ and identify the distribution of $X^2$ using the moment generating function. E[$e^{tx^2}]=\int_{-\infty}^{\infty}{e^{tx^2}e^{-x^2}\over{\sqrt{2\pi}}}$ which reduces to =$1\over{\sqrt{2\pi}}$ $\int_{-\infty}^{\infty}{e^{tx^2}e^{-x^2}}$ =$1\over{\sqrt{2\pi}}$ $\int_{-\infty}^{\infty}{e^{x^2(t-\frac12)}}$ and thus I am stuck. I’m sure there must be some trick to this (like completely the square for the mgf of […]

Comparison of the change of variable theorem

I would like to compare the change of variable theorem for 1 variable and more. What are the differences, in which case we need stronger assumptions? How do they differ? What is the best way to write the theorems for comparison? Multivariable: Let $\varphi : \Omega \rightarrow \mathbb{R}^n$ (where $\Omega \subset \mathbb{R}^k$ and $k\leq n)$ […]