Articles of integration

Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$

How do I integrate this? $$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$ I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of problems about complex integration, I thought there must be another (easier?) way. I tried computing the poles in the complex plane and got $$\text{Re}(z_0)=\pi+2\pi k, k\in\mathbb{Z}; \text{Im}(z_0)=-\log (2\pm\sqrt{3})$$ but what […]

Differentiation under the double integral sign

Working on three-body dispersion forces I got the following quantity: $$\frac{\partial } {{\partial \lambda }}\int\limits_\lambda ^{\pi – \lambda } {d\theta } \int\limits_\lambda ^{\pi – \lambda } {d\phi f\left( {\theta ,\phi } \right)} $$ where f is independent of $\lambda$. My question is: how can i take the derivative under the double integral sign? In one-dimensional […]

Why is the hypothesis of continuous differentiability necessary for integration by substitution?

This is probably a dumb question but I don’t quite get why the hypothesis that $g’$ is continuous on $(a,b)$ is necessary for the theorem $$\int_a^b (f \circ g) \ g’ = \int_{g(a)}^{g(b)} f$$ to hold. Almost all of the sources I have seen assume this hypothesis, even though it is not used explicitly in […]

Double integral of an function odd with respect to $y$ over a domain symmetric with respect to $x$ axis

Let $D$ be a region given as the set of $(x, y)$ with $$a \leq x \leq b\quad\text{and}\quad-\Phi(x) \leq y \leq \Phi(x)$$ where $Φ$ is a nonnegative continuous function on the interval $[a, b]$. Let $f(x, y)$ be a function on $D$ such that $$f(x, y) = – f(x, – y)$$ for all $(x , […]

Riemann-Stieltjes integral of unbounded function

In many theorems about the Riemann-Stieltjes integral they required the hypothesis of $f$ to be bounded (for example: Suppose that $f$ is bounded in $[a,b]$, $f$ has only finitely many points of discontinuity in $I=[a,b]$, and that the monotonically increasing function $\alpha$ is continuous at each point of discontinuity of $f$, then $f$ is Riemann-Stieltjes […]

Can this function be expressed in terms of other well-known functions?

Consider the function $$f(a) = \int^1_0 \frac {t-1}{t^a-1}dt$$ Can this function be expressed in terms of ‘well-known’ functions for integer values of $a$? I know that it can be relatively simply evaluated for specific values of $a$ as long as they are integers or simple fractions, but I am looking for a more general formula. […]

Show that $f$ is identically zero if $|f(x)|\leq\int_0^xf(t)dt$

Suppose $f:[0,1]\to\mathbb R$ is a continuous function satisfying $$|f(x)|\leq\int_0^xf(t)dt$$ for all $x\in[0,1]$. Show that $f$ is identically zero. I note that $f(0)=0$ trivially. Then how should one proceed? Any hint is appreciated. I proceeded using the following but scrapped that as I am sure it’s wrong. Since $f$ is continuous on $[0,1]$ it must attain […]

Convolution of half-circle with inverse

I am trying to compute the function: $$f(\lambda)\equiv\int_{-1}^{1}\frac{\sqrt{1-x^2}}{\lambda-x}dx.$$ It arises as the convolution of the semi-circle density with the inverse function. When $\lambda\in(-1,1)$ it can only be defined as a Cauchy Principal Value. I have a hunch that I need to go into the complex plane to solve this, but am not sure how to […]

Evaluating a logarithmic integral in terms of trilogarithms

For $a,c\in\mathbb{R}\land-1\le a\land-1<c$, define the function $J{\left(a,c\right)}$ to be the value of the dilogarithmic integral $$J{\left(a,c\right)}:=\int_{0}^{1}\mathrm{d}y\,\frac{\operatorname{Li}_{2}{\left(\frac{c}{1+c}\right)}-\operatorname{Li}_{2}{\left(\frac{ay}{1+ay}\right)}}{c-ay}.$$ In principle, $J{\left(a,c\right)}$ may be evaluated in terms of trilogs, dilogs, and elementary functions. In the process of trying to develop my own solution, I managed to obtain partial solutions valid over various subsets of the parameter space $(a,c)\in[-1,\infty)\times(-1,\infty)$, […]

Multiple self-convolution of rectangular function – integral evaluation

I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise. I define $$ g_2 (y) = \int_{-\infty}^{\infty} f(y+x) f(x) \mathrm{d} x $$ and recursively $$ g_n (y) = \int_{-\infty}^{\infty} g_{n-1}(y+x) f(x) \mathrm{d} x \, . $$ To make it easier […]