Articles of integration

Integral of an increasing function is convex?

Let $f$ be a real valued differentiable function defined for all $x \geq a$. Consider a function F defined by $F(x) = \int_a^x f(t) dt$. If f is increasing on any interval, then on that interval F is convex. I am not sure I intuitively understand this. What is the function is increasing at an […]

Composition of functions question

Am restricting this question to the elementary context of Riemann integrals and continuous functions $f,g.$ Because this came up in the context of another question, I would prefer to keep the examples from that question, at the risk of artificiality. Let $g:[0,4]\to \mathbb{R}$ such that $$\int_0^1g(x)dx = \int_3^4 4-g(x)dx.~~~~~(1)$$ An example would be $g(x) = […]

How to find the integral $\int \tan (5x) \tan (3x) \tan(2x) \ dx $?

Find the integral $\int \tan (5x) \tan (3x) \tan(2x) \ dx $ . This question is posted in a maths group in Facebook. What way should we use to solve integral like this? Thanks in advance

A closed form for $\int x^nf(x)\mathrm{d}x$

When trying to find a closed form for the expression $$\int x^nf(x)\mathrm{d}x$$ in terms of integrals of $f(x)$ I found that $$\int xf(x)\mathrm{d}x=x\int f(x)\mathrm{d}x-\iint f(x)(\mathrm{d}x)^2$$ $$\int x^2f(x)\mathrm{d}x=x^2\int f(x)\mathrm{d}x-2x\iint f(x)(\mathrm{d}x)^2+2\iiint f(x)(\mathrm{d}x)^3$$ $$\int x^3f(x)\mathrm{d}x=x^3\int f(x)\mathrm{d}x-3x^2\iint f(x)(\mathrm{d}x)^2+6x\iiint f(x)(\mathrm{d}x)^3-6\iiiint f(x)(\mathrm{d}x)^4$$ In general, it seems to be the case that $$\int x^nf(x)\mathrm{d}x=\sum_{k=0}^n(-1)^k\frac{\mathrm{d}^k}{(\mathrm{d}x)^k}x^n\int^{k+1}f(x)(\mathrm{d}x)^{k+1}$$ Using a formula for the repeated derivative of […]

Volume using Triple Integrals

Find the volume of solid enclosed by surfaces $x^2+y^2=9$ and $x^2+z^2=9$ I understand that these are two cylinders in XY and XZ planes respectively, that will cut each other above the XY plane. I get the following limits for triple integrals $$\int_{\theta=0}^{2\pi}\int_{r=0}^3\int_{z=0}^{\sqrt{ 9-r^2\cos^2\theta}}rdzdrd\theta$$ Is it correct ?? If yes, the integral itself looks so complicates. […]

Mollifiers: Approximation

Problem Given a mollifier: $\varphi\in\mathcal{L}(\mathbb{R})$ Then it acts as an approximate identity: $$f\in\mathcal{C}(\mathbb{R}):\quad\int_{-\infty}^\infty n\varphi(nx)f(x)dx\to f(0)\cdot\int_{-\infty}^\infty\varphi(x)dx$$ How to prove this under reasonable assumptions? Example As an example regard the Gaussian: $$f\in\mathcal{C}(\mathbb{R}):\quad\frac{n}{\sqrt{\pi}}\int_{-\infty}^\infty e^{-(nx)^2}f(x)\mathrm{d}x\to f(0)$$ (This is a useful technique when studying operator semigroups.)

Derivative of a function is the equation of the tangent line?

So what exactly is a derivative? Is that the EQUATION of the line tangent to any point on a curve? So there are 2 equations? One for the actual curve, the other for the line tangent to some point on the curve? How can the equation of the tangent line be the same equation throughout […]

Using Parseval's theorem to solve an integral

The question at hand is to use Parseval’s theorem to solve the following integral: $$\int_{-\infty}^{\infty} sinc^4 (kt) dt$$ I understand Parseval’s theorem to be: $$E_g = \int_{-\infty}^{\infty} g^2(t) = \int_{-\infty}^{\infty} |G(f)|^2 df $$ I began by doing the obvious and removing the squared such that: $$g^2(t) = Sinc^4 (kt)$$ $$g(t) = Sinc^2 (kt)$$ Following the […]

Integral of $\int \frac {\sqrt {x^2 – 4}}{x} dx$

I am trying to find $$\int \frac {\sqrt {x^2 – 4}}{x} dx$$ I make $x = 2 \sec\theta$ $$\int \frac {\sqrt {4(\sec^2 \theta – 1)}}{x} dx$$ $$\int \frac {\sqrt {4\tan^2 \theta}}{x} dx$$ $$\int \frac {2\tan \theta}{x} dx$$ From here I am not too sure what to do but I know I shouldn’t have x. $$\int […]

how could we compute this infinite real integral using complex methods?

$\int^{\infty}_{-\infty} \frac{cos(x)}{x^4+1}dx$ I know a similar result, but I’m not sure if I can take it for granted, that $\int^{\infty}_{-\infty} \frac{cos(x)}{x^2+1}dx = \frac{\pi}{e}$ The section in the book is related to Cauchy’s Integral Formulas and Liuville’s th’m, but I’m not sure how to apply these here.