Dx Definition and 306 Threads

The Nikon DX format is an alternative name used by Nikon corporation for APS-C image sensor format being approximately 24x16 mm. Its dimensions are about 2⁄3 (29 mm vs 43 mm diagonal, approx.) those of the 35mm format. The format was created by Nikon for its digital SLR cameras, many of which are equipped with DX-sized sensors. DX format is very similar in size to sensors from Pentax, Sony and other camera manufacturers. All are referred to as APS-C, including the Canon cameras with a slightly smaller sensor.
Nikon has produced 23 lenses for the DX format, from macro to telephoto lenses. 35mm format lenses can also be used with DX format cameras, with additional advantages: less vignetting, less distortion and often better border sharpness. Disadvantages of 35mm lenses include generally higher weight and incompatible features such as autofocus with some lower-end DX cameras. Nikon has also produced digital SLRs that feature the larger Nikon FX format sensor that is the size of the 135 film format.
In 2013, Nikon introduced a high-end compact camera with a DX-sized sensor, the Nikon Coolpix A, featuring an 18.5 mm lens.

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  1. M

    MHB Integrating e^X - e^-x/e^-x+1 dx

    how do you integrate e^X-e^-x/e^-x+1 dx i am trying multiplying by e^x and trying to make it into no fraction but i am having no luck
  2. karush

    MHB 242.8.2.8 int x sin (x/5) dx. IBP

    $\large{242.8.2.8}$ $\displaystyle I_8=\int(x)\sin{\left(\frac{x}{5}\right)} \, dx= 25\sin\left(\dfrac{x}{5}\right)-5\cos\left(\dfrac{x}{5}\right)x$ $$\begin{align} u&=\frac{x}{5} &5du&=dx &x&=5u \\ \end{align}\\ $$ thus $\displaystyle I_8=25\int u\sin{u} \, du$ IBP $$\begin{align} u_1&=u...
  3. Consolacion Ruiz

    I What is the difference between dx, Δx and δx?

    What is the difference between dx, Δx and δx? Δ = difference d = Δ but small difference, infinitesimal δ = d but along a curve Mathematical symbols are always graphics.I’m not sure if that will be true, but it would be beautiful.
  4. karush

    MHB Thanks! Glad you found it helpful.

    Whitman 8.3.12 $$\int \frac{{x}^{3 }}{\sqrt{4x ^2 - 1}} \ dx = \frac{\left(2{x}^{2}+1\right)\sqrt{4{x}^{2}-1}}{24}$$ $$u=4x^2 - 1 \ \ \ \ du=8x \ dx \ \ \ x=\left(\frac{u-1}{4 }\right)^\frac{1}{2}$$ Substitute and simplify $$\frac{1}{32}\displaystyle \int\dfrac{u+1}{\sqrt{u}}\,\mathrm{d}u$$
  5. karush

    MHB -w8.3.11 int sqrt{x} dx sqrt{1-x}

    w8.3.11 nmh{1000} $\displaystyle I= \int\frac {\sqrt{x}} {\sqrt{1-x}}\ dx=\arcsin\left({\sqrt{x}}\right)-\sqrt{x}\sqrt{1 - x}+C $ Substitutions $x=\sin^2 \left({u}\right) \quad dx=2\sin\left({u}\right) \cos\left({u}\right) \ du \quad u=\arcsin\left({\sqrt{x}}\right)$ This evaluates to...
  6. karush

    MHB How Do You Integrate 1/(x^2 * (1 + x^2))?

    Whitman 8.3.9 $$\displaystyle \int\frac{1 }{{x}^{2}\left(1+{x}^{2}\right)} \ dx =-\arctan\left({x}\right)-\frac{1}{x}+C $$ Expand $$\displaystyle \int\frac{1}{{x}^{2}}\ dx -\int \frac{1}{{x}^{2}+1}dx $$ Solving $$\displaystyle \int\frac{1}{{x}^{2}}\ dx =-\frac{1}{x}+C$$ Solving...
  7. karush

    MHB How can I use integration by parts to solve $\displaystyle \int\sin^2(x) \ dx$?

    mnt{w.8.4.5} nmh{1000} $\displaystyle \int\sin^2 \left({x}\right) \ dx = \frac{x}{2}-\frac{\sin\left({2x }\right)}{4}+C $ As given by a table reference Integral calculator uses reduction formula to solve this But this is an exercise following integration by parts so.. $\displaystyle \int\sin^2...
  8. karush

    MHB 8.4.2 - Computing ∫ x² cos(x) dx

    8.4.2 $$\int {x}^{2}\cos\left({x}\right)\ dx = {x}^{2}\sin\left({x}\right) -2\sin\left({x}\right) +2x\cos\left({x}\right) + C $$ $$uv-\int v \ du $$ $$u={x}^{2}\ \ \ dv=\cos\left({x}\right)\ dx $$ $$du=2x \ dx \ \ \ \ v=\sin\left({x}\right)$$...
  9. karush

    MHB How Do You Evaluate ∫x²√(1-x²) dx Using Trig Substitution?

    W8.3.6 evaluate $$\int {x}^{2}\sqrt{1-{x}^{2 }} \ dx = \arcsin\left({x}\right)/8—\sin\left({4\arcsin\left({x}\right)}\right)/32 + C $$ This is from an exercise on trig substitutions so $$x=\sin\left({x}\right) \text{ so } \int\sin^2 \left({x}\right)\sqrt{1-\sin^2 \left({x}\right)}\ dx...
  10. karush

    MHB What is the integral of $\sqrt{x^2-1}$? What is the integral of $\sqrt{x^2-1}$?

    Evaluate $$\displaystyle \int\sqrt{{x}^{2}-1} \ dx$$ First the indenitly of $\tan^2 \left({x}\right)=\sec^2 \left({x}\right)-1$ fits the expression in the radical But not sure how to set up the substitution
  11. karush

    MHB -z.61.W8.6 int sin^2(x)cos^2(x) dx

    Evaluate $\displaystyle \int\sin^2 \left({x}\right)\cos^2 \left({x}\right)$ $\displaystyle \sin^2\left({x}\right) =\frac{1-\cos\left({2x}\right)}{2}$ and $\displaystyle \cos^2 \left({x}\right) =\frac{1+\cos\left({2x}\right)}{2}$ So $\displaystyle \int\frac{1-\cos\left({2x}\right)}{2}...
  12. P

    Show that ∫ dx |f(x)|^2 = ∑ |Cn|^2

    Homework Statement This is a question regarding Fourier series. ∫ dx |f(x)|^2 = ∑ |Cn|2 (note the integral is between -π and π, and the sum is from n= -∞ to ∞)Homework Equations Complex Fourier series: f(x) = ∑Cn einx (again between n = -∞ and ∞) The Attempt at a Solution So I figured the...
  13. N

    I How Do You Solve the Integral of 1/(y+cos(x))^2?

    First part of the question was to work out the integral 1/(y+cos(x)) between x=0 and x=pi/2 by using the substitution t=tan(x/2). Got this to be \frac{2}{\sqrt{y^2-1}}arctan(\sqrt{\frac{y-1}{y+1}}) The next question says HENCE find integral with the same limits of \frac{1}{(y+cos(x))^2} Ive...
  14. S

    Calculating Integral: ##\int_a^b x\left(\frac{b-x}{b-a}\right)^{n-1} \; dx##

    Homework Statement How do I calculate ##\int_a^b x\left(\frac{b-x}{b-a}\right)^{n-1} \; dx##? Homework EquationsThe Attempt at a Solution I tried the substitution ##u = \frac{b-x}{b-a}## to no avail. Someone please help.
  15. karush

    MHB How to Integrate cos^2(x)sin^3(x) dx?

    $$\int \cos^2 \left({x}\right)\sin^3 \left({x}\right)dx$$ $$-\int\cos^2 \left({x}\right)\left(1-\cos^2 \left({x}\right)\right)\sin\left({x}\right)dx =-\int\left(\cos^2 \left({x}\right)-\cos^4\left({x}\right)\right)\sin\left({x}\right)dx $$ $$u=\cos\left({x}\right)\ \ du=-\sin\left({x}\right)dx...
  16. H

    How Do You Integrate tan²x sec^4x dx?

    Homework Statement the correct solution is ∫ tan²x sec²x sec²x dx = replace the first sec²x with (tan²x + 1): ∫ tan²x (tan²x + 1) sec²x dx = expand it into: ∫ (tan^4x + tan²x) sec²x dx = let tanx = u differentiate both sides: d(tanx) = du → sec²x dx = du substituting, you...
  17. A

    How Do You Integrate ##\int \tan 2x \ dx##?

    What is ##\int \tan 2x \ dx##? What I get is ##\int \tan 2x \ dx## ##= \int \frac{\sin 2x}{\cos 2x} dx## ##= \int \frac{2 \sin x \cos x}{1 - 2 \sin^2 x}dx## let u = sin x then ##\frac{du}{dx} = \cos x## or du = cos x dx So ##= \int \frac{2 \sin x \cos x}{1 - 2 \sin^2 x}dx## ##= \int...
  18. 1

    What Happens to dx in Integration?

    Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually. When integrating dy/dx=x you get, ∫x dx=1/2x2. But what exactly happened to the dx, why did it...
  19. Y

    Thermodynamics: Relationship between deltaX, partialX, dx

    Homework Statement I am trying to understand the the following derivation: Cv = (qv/ΔT) = (ΔU/ΔT) \\ Cv = (∂U/∂T)v \\ dU = CvdT The Attempt at a Solution [/B] So here is what I understand so far. I understand that heat transfer q and temperature T are related by a direct...
  20. Eclair_de_XII

    Can you derive a trigonometric function from its inverse dx?

    Homework Statement Arbitrary derivative of inverse trigonometric function: (sin-1x) = 1/(√1 - x2) Homework Equations f-1(f(x)) = 1/f`(x) The Attempt at a Solution So basically I learned about derivatives of trigonometric functions in class, and I thought maybe this would work: deriving the...
  21. Kelly333

    Why can "dx" in integration be multiplied?

    Hi, This is an example in "Barron AP calculus" I learned from some past threads that "dx" in integration either means △x which is a infinite number or indicates the variable with respect to which you're integrating. In the equation above, it seems that dx is multiplied by (1-3x)^2. Isn't dx...
  22. Oribe Yasuna

    Integrating dx / (4+x^2)^2 using Trigonometric Substitution

    Homework Statement Evaluate the integral: integral of dx / (4+x^2)^2 Homework Equations x = a tan x theta a^2 + x^2 = a^2 sec^2 theta The Attempt at a Solution x = 2 tan theta dx = 2sec^2 theta tan theta = x/2 integral of dx / (4+x^2)^2 = 1/8 integral (sec^2 theta / sec^4 theta) d theta =...
  23. karush

    MHB -z.57 dx formula that estimates the change

    Write a differential formula that estimates the change in the volume $V=\frac{4}{3}\pi{r}^{3}$ of a sphere when the radius changes from $r_0$ to $r_0+dr$ $$dV=4\pi{r}_{0}^{2}dr$$ this was one of the selections but I didn't know how to account for the $r_0$ to $r_0+dr$
  24. Y

    Difficulty understanding ∫ P(X).X^2 dx = <X^2> ?

    I know for discrete random variables Σ P(x).x = <x> Translating for continuous random variables I'm also aware of the result ∫ P(x).x dx In my lecture notes ( I more or less transcribed from what the lecturer said ): ∫ P(x).x^2 dx = <x^2> , should it not be ∫ P(x^2).x^2 dx = <x^2>? Does...
  25. B

    Definition of dx: What is its Domain & Formalization?

    Homework Statement http://imgur.com/goozE9f Homework Equations ##(dx_i)_p i= 1,2,3## 3. The Attempt at a Solution [/B] I'm reading Manfredo and Do Carmo's Differential Forms and Applications. This is the very first page Would you mind explaining me what is meant by dx, as highlighted in the...
  26. D

    How Do You Integrate (dx/dt) dx in Physics Problems?

    Good Night, Can someone please tell me how to do: ∫ b (dx/dt) ⋅ dx ? Like in the work done by a force which is proportional to the velocity (like drag). I tried to change dx to v dt but couldn´t go much further. Thank you in advance.
  27. B

    MHB Stumped by Definite Integral: $\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx$

    Was asked to solve this definite integral in a tech free test. Not sure how to go about it. $$\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx.$$ I know here is a relationship between inverse sin and the sqrt function but with just sin x?
  28. Stephanus

    Relativity: Exploring dX and dY in Math

    Dear PF Forum, I'm studying Relativity, And there's some term in math that I don't know. And I am even not an English native. in ##\frac{dX}{dY}## What does "d" mean? I know that dX is the tiny slice of X. What do we say it in English?
  29. A

    Solving (x+1)^2 dx for Integration - Step by Step Guide

    Hi, I literally just registered so I have no idea about forum rules, also I'm not good in english. 1. Homework Statement The equation is (x + 1)^2 dx. U = (x+1) DU = 1DX Homework EquationsThe Attempt at a Solution Here I get (U^3) over 3 times DX = (x^3 + 3x^2 + 3x + 1) over 3 times 1 I...
  30. Byeonggon Lee

    ##\int\frac{2x+6}{(x-1)(x+1)^2} dx## ?

    Hi I'm currently doing 'integral by substitution' part in a book. Although it is integral by substitution part, some exercises are solved using reduction of fraction and integral, without substitution. (Actually I can't solve some exercises if I use substitution and the book's explanation also...
  31. B

    Integrating dx and dy: What Does It Mean?

    Once you've integrated, dx and dy just indicate what variable you've integrated in terms of, correct?
  32. S

    Differential of X [ dX = U/V dV + U/p dp ] Internal Energy?

    Homework Statement dX = U/V dV + U/p dp Write the differential of X in terms of the independent variables.Prove that this is an exact differential.Use the ideal gas equation of state to verify that X is actually the internal energy and that it satisfies the above equation. Would...
  33. J

    Why Does My Integral Calculation Differ from the Expected Result?

    The result of the integral of (1/2)sin2x dx with: upper limit x = arccos((R-h)/R) lower limit x = 0 is (-h^2+2Rh)/(2R^2) I can not seem to get this exact answer my workings yield: let u = 2x, du/dx = 2 therefore dx = du/2 Integral becomes (1/4) ∫ sinu du (with the same upper and lower...
  34. Emmanuel_Euler

    What is the integral of e^-(x/2) * sin(a*x) dx?

    what is the integral of e^-(x/2) * sin(a*x) dx??
  35. Emmanuel_Euler

    Indefinite and definite integral of e^sin(x) dx

    Look to this indefinite integral →∫e^(sin(x))dx Antiderivative or integral could not be found.and impossible to solve. Look to this definite integral ∫e^(sin(x))dx (Upper bound is π and Lower bound is zero)=?? my question is : can we find any solution for this integral (definite integral) ??
  36. D

    What is the interpretation of dx in calculus?

    Apologies if this isn't quite the right forum to post this in, but I was unsure between this and the calculus forum. Something that has always bothered me since first learning calculus is how to interpret dx, essentially, what does it "mean"? I understand that it doesn't make sense to consider...
  37. karush

    MHB Integrating $\int x^2\cos\left({\frac{x}{2}}\right)dx$ by parts

    $\int x^2\cos\left({\frac{x}{2}}\right)dx$ $u={x}^{2}\ dv=\cos{\left(\frac{x}{2}\right)}dx$ $du=2x dx\ v=\int\cos\left({\frac{x}{2}}\right)dx=2\sin\left({\frac{x}{2}}\right)$ Integrat by parts, just seeing if getting started ok
  38. Ray Beaver

    Integrate x^2 / (x^2 + a^2)^3/2 dx

    Homework Statement My solution has two terms divided by a which is in error. I am a volunteer math coach to some junior college students and can't find my error in the problem. Its been a while since I earned my masters in electrical engineering. The issue is the presencce of the variable...
  39. genxium

    How to Determine Frame Size with Lorentz Transformation

    First time posting in this section. I understand that this question could possibly be an old and common question about Lorentz Transformation, however I failed to find useful discussions or instructions online. Assuming that there're 2 frames ##S, S'## where ##S'## moves along the ##x_{+}##...
  40. karush

    MHB How Can I Use Substitution to Solve This Integral?

    $$\int\frac{2x}{x^2+9}\ \text{dx}$$ I thot I could use $$u={x}^{2}+9$$ But counldn't go thru with it
  41. G

    How to Handle Integral with dx Inside the Function?

    Im working on a problem whit the follewing integral: I = \int|f(x+dx)dx| Im trying to use int by parts : t = x + dx \Rightarrow dt/dx = 1 + ddx/dx = ?, but i have no idee on what ddx/dx is? I think dx -> konstant? so ddx/dx = 1 ?
  42. S

    Compute ∫√(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum

    Homework Statement Integrate √(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum Homework Equations lim n→∞ Σ_(i=1)^n i = n(n+1)/2 lim n→∞ Σ_(i=1)^n i^2 = n(n+1)(2n+1)/6 The Attempt at a Solution Δx = (b - a)/n Δx = (5 - 0)/n Δx = 5/n f(x_i) = √(25 - [a + iΔx]^2) f(x_i) = √(25 - [0 +...
  43. karush

    MHB How Do You Integrate $\int \frac{1}{1-e^x} dx$ with Substitution?

    $\int \frac{1}{1-e^x} dx$ $u=1-e^x,\ du=-e^x dx$ Not sure of next step
  44. karush

    MHB What is the Integral of e^√x/√x?

    $$\int\frac{e^{\sqrt{x}}}{\sqrt{x}}dx$$ ok I set $u=\sqrt{x}$ and $du=\frac{1}{2\sqrt{x}}dx$ I thot I would find a table reference for this but not sure which one could be used so now we have $$\frac{1}{2}\int\frac{e^{u}}{u}du$$ but maybe better way
  45. johann1301

    Integrating cos^2x with the Chain Rule: Explanation and Example

    Homework Statement ∫cos2x dx The Attempt at a Solution I know the answer, and i know how to get there using: cos2x+sin2x=1 cos2x-sin2x=cos2x cos2x=(1+cos2x)/2 But why can't i use the chain rule? Can i?
  46. W

    Integral x^n *f(x) dx =0 ; f for all n, f in C[0,1], then f(x)=0

    I am trying to show that for f in C[0,1] , and ##n=0,1,2,... ## we have: ## \int_0^1 x^n f(x)dx =0 ## (&&) , then ##f(x)==0 ## . I am using Weirstrass approximation, so that , for any ## \epsilon >0 ## , there is ## P_n(x) = a_0+a_1x +..+x^n ## with : ##Sup_{x in [0,1]} |...
  47. Rochefort

    Trapezium rule, what does dx represent

    Homework Statement What does the "dx", associated with the definite integral represent for the trapezium rule? Could dx=h? (the heights of the trapeziums) Homework Equations The Attempt at a Solution
  48. C

    Integrating tan^3 x: Tips and Tricks for Solving ∫tan3x dx

    Homework Statement ∫tan3x dx 2. The attempt at a solution ∫tan x + ∫tan2x ∫tan x (sec2x - 1) dx ∫(tan x (sec2x - tan x) dx ∫tan x sec2x dx - ∫tan x dx u = sec x du = sec x tan x dx ∫tan x sec x sec x - ∫tan x dx Now I'm stuck.. ∫ du * u - ∫ tan x dx ?
  49. M

    Finding Integrals: ∫ (5x^2 + sqrt(x) - 4/x^2) dx

    I have these integrals to find: ∫ (5x^2 + sqrt(x) - 4/x^2) dx ∫ [cos(x/2) - sin(3x/2)] dx ∫ s/sqrt(s^2 + 4) ds (upper coordinate is 5 lower coordinate is 1) I have worked it out as: ∫〖(5x^2+√x〗-4/x^2) dx=5x^3/(2+1)+x^(1/2+1)/(1+1/2)-4x^(-2+1)/(-2+1)+C=5/3 x^3+2/3x^(3/2)+4/x+C...
  50. KingCrimson

    Why do we write dx in indefinite integrals

    in understand why we write the dx in riemann integral , but in the indefinite integral why do we use that ? what is the relation between the area under a curve , and the antiderivative of that of that curve ??
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