Dx Definition and 306 Threads

Anyone have some ideas to approach the integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##?

still typing...checking latex

What is ##\int \frac{1}{x^2 + 3} \ dx##? This is my attempt: ##x = \sqrt{3} \tan \theta## --> ##dx = \sqrt{3} \sec^2 \theta \ d\theta## ##x^2 + 3 = (\sqrt{3} \tan \theta)^2 + 3## ##= 3 \tan^2 \theta + 3## ##= 3 (\tan^2 \theta + 1)## ##= 3 \sec^2\theta## ##\int \frac{1}{x^2 + 3} \ dx = \int...

Hello! As is known, \Delta y = dy for infinitesimally small dx. It's true. But if we have graph we may see that \Delta y isn't equal to dy even for infinitesimally small dx. Why is that so? Thanks!

Hi, Is it correct to say that the dirac delta function is equal to 0 except if the argument is 0? Thus, ##x^2 +2x -3## must be equal to 0. Then, we have x = 1 or -3. What does that means? ##\int_{-\infty}^{\infty} e^{-|x|}\delta(x^2 +2x -3) dx = e^{-1}## and/or ##e^{-3}## ? Thank you

Find $\dfrac{dy}{dx}$ for:$y=\dfrac{7+9x-6\sqrt{x}}{x}$ ok several ways to solve this but thot the best might be to expand it first so $y=\dfrac{7}{x} +9 -6\dfrac{\sqrt{x}}{x}$ or is there a better way

This is the question ... I have it's solution ... My problem : I can't understand why dx=R/cos^2(teta) dteta I have thought many hours but I couldn't find it's reason ... Can anyone please help with this ?!

I can calculate the divergence $$div A = 2y + 2x + z + 8z^3$$ Now I have to integrate over this cut-off sphere. So I decide I'll cut it up into small discs with height dy and integrate over that $$dV = \pi(4 - (y-1)^2)^2 * dy$$ My issue here is I don't know how to integrate 2x + z + 8z^3. Not...

Reif says " ... variable ##u## which can assume any value in the continuous range ##a_{1}<u<a_{2}##. To give a probability description of such a situation, one can focus attention on any infinitesimal range of the variable between ##u## and ##u+d u## and ask for the probability that the variable...

Evaluate $\displaystyle\int_0^\infty e^{-\dfrac{x^2}{2}} dx$ ok first reponse is use IBP but can we use $e^u$ where $u=-\dfrac{x^2}{2}$ ot $u=\dfrac{x}{\sqrt{2}}$

Evaluate $I=\displaystyle \int 64x\sec^2(4x) \, dx$ ok well first $64 \displaystyle\int x\sec^2(4x) \, dx$ off hand not sure what trig id to use or if we need to

What's the difference between d,d/dx and dx?

Mentor note: Moved from technical section, so missing the homework template. How do you integrate this? $$\int \frac{1}{x^2 + 2} dx$$ My attempt is $$\ln |x^2 + 2| + C$$

How do you integrate ##\frac{1}{\sqrt{x^3 + x^2 + x + 1}} \, dx##? Please give me some hints and clues. Thank you

How do you integrate ##\frac{1}{\sqrt{x^2 + 3x + 2}} dx##? I had tried using ##u = x^2 + 3x + 2## and trigonometry substitution but failed. Please give me some clues and hints. Thank you mentor note: moved from a non-homework to here hence no template.

Good Morning To cut the chase, what is the dx in an integral? I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx That said, I have seen it in an integral, specifically for calculating work. I do understand the idea of...

Here is the picture on the system. I have to find the period (T). The masses, R and dX is given. The systam at first is at rest, then at t = 0 we pull the plank to dX distance from its originial position. In the thread...

This is more of a "housekeeping" question, though I haven't studied much in the way of infinitesimals so apologies in advance for my lack of rigour! As far as I'm aware, an infinitesimal can be thought of as a small change in some quantity. Changes can be either positive or negative, so...

As a simple example, the probability of measuring the position between x and x + dx is |\psi(x)|^{2} dx since |\psi(x)|^{2} is the probability density. So summing |\psi(x)|^{2} dx between any two points within the boundaries yields the required probability. The integral I'm confused about is...

7.2.6a use $\tan^2 x= \sec^2 x-1$ to evaluate $$\displaystyle I_{6a}=\int\tan^4 x \, dx$$ well my first inclinations is to. $$\int(\sec^2 x-1)^2\, dx$$ then expand $$\int (\sec^4 x - 2 \sec^2 x +1)$$ ok not sure if this is the right direction W|A returned this:

$\tiny{213(DOY)}$ $\displaystyle\int \sec x \tan x \: dx =$ (A) $\sec x + C$ (B) $\tan x + C$ (C) $\dfrac{\sec^2 x}{2}+ C$ (D) $\dfrac {tan^2 x}{2}+C $ (E) $\dfrac{\sec^2 x \tan^2 x }{2}+ C$

In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))... ## \int_0^\phi \frac {d} {dx} f(x) dx =...

U= 1/x dV= sin(x) dU = -1/x^2dx V= -cos(x). lim b--> infiniti (integral from [0,b]) = 1/x(-cos(x)) - integral(1/x^2(cos(x)) dx

Hello! Let ##I## be an interval of size ##L##, suppose we divide it into bits of ##dx## then ##L=dx+dx+... =\alpha.dx## Since ##dx## is by definition infinitesimally small is it correct to imply that for each ##x## there's a corresponding ##dx## hence, here, ##\alpha## would be, theoretically...

Homework Statement ∫1/(√3 sinx+ cosx) dx is Ans. as per textbook 1/2(log(tan((x/2)+(pi/12)) 2. The attempt at a solution Attempt 1-I changed it to half angles (x/2) and then multiplied and divided sec^2(x/2) to the numerator and denominator, then putting tan(x/2)=t I got 2∫1/(√3t+1-t^2) I wrote...

Homework Statement What does 'dv and dx' mean in f = eta A (dv/dx) in coefficient of viscosity? 2. Homework Equations What does 'dv and dx' mean in f = eta A (dv/dx) in coefficient of viscosity? The Attempt at a Solution As I know v = velocity and x = distance. then what does 'dv' and 'dx'...

Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?

What is dx, dy and dz in spherical coordinates

Let us look at short segment of a rod with its length dx. Due to longitudinal wave, left endpoint moves for s in the direction of x-axis and the right endpoint moves in the same direction for s+ds. Because I want to calculate the elastic energy of the wave motion, I need the extension of dx so...

Homework Statement This isn't really a homework problem/question. I am just wondering if it is possible to calculate the vertical distance (dy) when the only given values are those of the horizontal distance (dx) and velocity (v). An example of this projectile-motion problem would be this: A...

Can dx be thought of as a sufficiently small change in x? I want to say that dx is the change in x and change in x approaches 0, but that would just be 0. So I think it might make more sense to just say sufficiently small. Then when we look at something like a derivative dy/dx we can look at...

Often I have to solve problems using dx or dq. I always don’t quite understand what’s going on. I understand what dy/dx is but not just dx. Can someone walk me through in plain language a somewhat rigorous definition of differential like dx?

When solving physics problems sometimes we have to use differentials like dx or dq. I don’t quite understand how to use these. I understand that the limit as change in x approaches 0 of change in y over change in x is represented by dy/dx, where dy and dx are sometimes said to be small changes...

I previously made a derivation of Neumann potential. It can be found in the pdf file below. I originally made it in the ##dx## method. It involved equations like ##dm=I dS##. My maths teacher told that such an expression has no meaning, at least in elementary calculus. However I argued that my...

Hi, todos: Do you know how to calculate the definte integral for Integral for ##\int_{-1}^{1} [P_{l}^{m}]^2 \ln [P_{l}^{m}]^2 dx##, where ##P_{l}^{m} (x)## is associated Legendre functions. Thanks for your time and help.

This is the continuation of the below thread: https://www.physicsforums.com/threads/what-is-integral-tan-2x-dx.856530/ Can someone please tell me how to integrate tan 2x dx?

Hi everyone I am reading Sean Carrol's lecture notes on general relativity. link to lecture : https://arxiv.org/abs/gr-qc/9712019 In his lecture he introduced dxμ as the coordinate basis of 1 form and ∂μ as the basis of vectors. I understand why ∂μ could be the basis of the vectors but not for...

$\tiny{8.3.8}$ $\textit{evaluate}$ \begin{align*}\displaystyle I_{8}&=\int_{0}^{\pi/4} \sin^5{2x},dx\\ &=-\int_{0}^{\pi/4} (\sin^2(2x))^2 sin(2x) \, dx \end{align*} $\textit{set $u=\cos{2x} \therefore du=-2\sin{(2x)} \, dx$ then $u(0)=1$ and $\displaystyle u(\pi/4)=0$}$...

Hello, I am slightly confused about the actual meaning of dx. Because I read in a physics textbook, they say something along this line: "We divide this region into spherical shells of radius r, surface area 4pr 2, thickness dr, and volume dV = 4pr^2 dr." I don't understand how can we represent...

$\tiny{242.7x.25}$ $\textsf{Find the derivative of y with respect to x}$ \begin{align*}\displaystyle y&=8\ln{x}+\sqrt{1-x^2}\arccos{x} \\ &=\frac{8}{x}+? \end{align*}

Hi all , I see this integral too much in QFT books when making loop calculations : ## \int_{0}^{1}~ dx~ dy~ dz~ \delta (x+y+z-1) = \int_{0}^{1} dz \int_{0}^{1-z} dy ## Can anyone explain how did we get this ? I mean it's apparent that ##\int_{0}^{1}~ dx \delta (x+y+z-1) ## have been...

Homework Statement I am doing a little review and having a some trouble deriving the integral ##\int \frac{1}{\sqrt{1-x^2}} dx## Homework EquationsThe Attempt at a Solution Initially I was trying to solve this integral using the substitution ##\cos \theta = x##. I drew my triangle so that...

Question: sqrt(x) cos(sqrt(x)) dx My try: Let dv = cos(√x) => v = 2√xsin(√x) and u = √x => du = dx/(2√x) Using integration by parts, we get ∫√x cos(√x) dx = 2√x√x sin(√x) - ∫(2√xsin(√x) dx)/(2√x) = 2x sin(√x) - ∫sin(√x) dx = 2x sin(√x) + 2 cos(√x) √x However, the answer given in the book...

Hi everyone, Can you tell me how to integrate the following equation? Integrate(1/(x(1+x^2)^0.5) dx

206.8.7.47 $\text{use Simpsons rule} \\ \text{n=8} $ $$\displaystyle \int_{0}^{3\pi/5} \sin\left({10x}\right)\cos\left({5x}\right)\,dx \approx \frac{4}{15}=0.2667$$ $$\displaystyle n=8\therefore \varDelta{x} =\frac{3\pi}{40} \\ S_{47}=\frac{\pi}{40}\left[ y_0+4y_1+2y_2 +4y_3+2y_4+4y_5...

$\text{206.8.7.64}$ $\text{Given and evaluation}$ $$\displaystyle I_{64}=\int \frac{x^4+1}{x^3+9x} \, dx =\dfrac{\ln\left(\left|x\right|\right)}{9}-\dfrac{41\ln\left(x^2+9\right)}{9}+\dfrac{x^2}{2}$$ $\text{expand (via TI)}$ $$I_{64}= \frac{1}{9}\int\frac{1 }{x} \, dx...

$\text{206.8.7.58}$ $\text{given and evaluation}$ $$\displaystyle I_{58}=\int \frac{dx}{{x}^{2}-6x+34} =\dfrac{\arctan\left(\frac{x-3}{5}\right)}{5} + C$$ $\text{complete the square} $ $${x}^{2}-6x+34 = \left(x-3\right)^2 + 5^2 = {u}^{2}+{a}^{2} \\ u=x-3 \\ a=5$$ $\text{standard integral} $...

$\text{206.8.7.32}$ Given and evaluation $$\displaystyle I_{32}=\int \tan^9\left({x}\right)\sec^4(x) \, dx =\dfrac{\tan^{12}\left(x\right)}{12} +\dfrac{\tan^{10}\left(x\right)}{10} + C$$ use identity $\tan^2\left({x}\right)+1=\sec^2\left(x\right)$ $$u=\tan\left(x\right) \therefore du...

$\tiny{206.8.4.61 \ calculated \ by \ Ti-nspire \ cx \ cas}$ $$I_{61}=\displaystyle \int\frac{x^2+2x+4}{\sqrt{x^2-4x}} \, dx =14\ln\left[{\sqrt{{x}^{2}-4x}}+x-2\right] +\left[\frac{x}{2}+5\right] \sqrt{{x}^{2}-4x}+C$$ $\text{complete the square}$ $$x^2-4x \implies \left[x-2\right]^2-4$$...

206.8.4.30. Int 24/(144x^2+1)^2 dx 206.8.4.30 $\displaystyle I_30=\int \frac{24}{(144x^2+1)^2}= \arctan\left(12x\right)+\dfrac{12x}{144x^2+1}+C$ So $x=12\tan\left({u}\right) \therefore du=12\sec^2 (u)du$ By the answer assume a trig subst. Didn't want to try reduction formula: Continue or is...