Dx Definition and 305 Threads

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

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

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.

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 $$\frac{1}{32}\displaystyle \int\dfrac{u+1}{\sqrt{u}}\,\mathrm{d}u$$

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...

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...

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...

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

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...

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

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}...

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...

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...

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.

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

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...

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...

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...

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...

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...

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...

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 =...

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 $r_0+dr$

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...

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...

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.

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?

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?

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...

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...

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

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...

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...

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

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...

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

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...

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_{+}##...

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

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 ?

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 +...

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

$$\int\frac{e^{\sqrt{x}}}{\sqrt{x}}dx$$ 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

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?

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]} |...

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

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 ?

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...

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 ??