Articles of integration

Drawing large rectangle under curve

Let $f$ be a continuous nonincreasing function on $[0,1]$ with $f(1)=0$ and $\int_0^1 f(x)dx=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k$, with sides parallel to the axes, in the area bounded by the two axes and the curve $f$? If we choose the […]

Integral $\int_1^2 \frac1x dx$ with a Riemann sum.

How do you find the $$ \int \dfrac{1}{x} dx$$ by using the idea of a limit of a Riemann sum on the interval [1,2]? I tried splitting the interval into a geometric progression and evaluating the Riemann sum, but i cant simplify the expression at this stage.

Convergence of integral $\int_{0}^{\infty}{\frac{1}{x^{3}-1}}dx$

So i have the integral $$\int_{0}^{\infty}{\frac{1}{x^{3}-1}} dx$$ Software programs say that it is divergent, except for one program which evaluate numerically and gave a clear result. The integrals $\int_{0}^{1}$ and $\int_{1}^{\infty}$ are indeed divergent if taken separately, but the fact that one goes to infinity and the other to minus infinity, makes me think that […]

Is this a solution to the indefinite integral of $e^{-x^2}$?

$\int e^{-x^2} \, \mathbb{d} x$, the Gaussian integral, is notorious throughout physics and statistics. Its definite integral defined over $\mathbb{R}$ is $\sqrt{\pi}$. However, the current indefinite integral is not an elementary solution. Recently I thought about an integration technique that uses “error correcting.” If we define a function $f(x)$ such that $f^{(n)}(x)$ still contains terms […]

Algorithm/Procedure for finding $\sigma$ such that $\omega=d\sigma$

I know that the Poincare’s lemma asserts that under certain conditions a differential form $\omega$ is exact, i.e. it possesses an antiderivative $\sigma$, such that $\omega=d\sigma$. But as far as I can tell, the proof does not contain an algorithm for actually finding such an $\sigma$ (at least in the way we sketched the proof […]

Fourier Transform calculation

I am trying to calculate the Fourier Transform of $$f(x)=\exp(-\frac{|x|^2}{2}). $$ Thus, I am looking at the integral $$ \hat{f}(u)=\int_{\mathbb{R}^n} \exp(-\frac{|x|^2}{2}) \cdot \exp(ix\cdot u) dx. $$ I can’t figure out how to evaluate this integral. Am I trying the wrong approach to calculate the transform or should I be able the integral. Note the integral […]

A primitive function of $ e^{x^{2}} $

I made some efforts to set a closed form of primitive function of $ e^{x^{2}} $ i find this function : $ f(x)=\frac{x}{2x^{2}-1}e^{x^{2}} $ where : $f'(x)=(\frac{x}{2x^{2}-1}e^{x^{2}})’$= $\frac{4x^{4}-4x^{2}-1}{4x^{4}-4x^{2}+1}e^{x^{2}}$ my question is : can I take $ f(x)=\frac{x}{2x^{2}-1}e^{x^{2}} $ as a primitive function of $ e^{x^{2}} $ if $x$ hold a big values ? is there […]

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ \int_{-\infty}^{\infty}\exp\left(-t^{2}\right)\,\mathrm{erfc}\left(t-c\right)\,\mathrm{erfc}\left(d-t\right)\,\mathrm{d}t=\frac{4}{\pi}\int_{-\infty}^{\infty}\exp\left(-t^{2}\right)\int_{t-c}^{\infty}\int_{d-t}^{\infty}\exp\left(-u^{2}-v^{2}\right)\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}t $$ So I note that $u\geq t-c$, $v\geq d-t$, thus $t\leq u+c$ and $t\geq d-v$, thus $d-v\leq t\leq u+c$ and $u+v\geq […]

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates. First of all I tried to find the domain of $x$ and $y$: $0 \le y \le x$ (given) $0 \le x^2 + y^2 \le […]

$\lim_{\lambda \to \infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t} $

For a continuous function, $f:[0,b] \to \Bbb{R}$ show that: $$ \lim_{\lambda\to\infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t}\,dt = \frac{\pi}{2}\,f(0) $$ I know it has something to do with the Riemann-Lebesgue lemma about Fourier series, but $\frac{f(t)}{t}$ is not an integrable function. I tried to define $g(t) = \frac{2f(t)\sin(\frac{t}{2})}{t} $ which tends to $f(0)$ as $t\to0$ and $g(t)$ […]