Integrating Definition and 940 Threads

Homework Statement \int \frac{t^{3}}{\sqrt{3 + t^{2}}} Homework Equations ∫udv = uv - ∫vdu The Attempt at a Solution So I tried integration by parts, then I had to integrate the last term with the same method, and then I got a u substitution integral, in the end I got. \int...

Homework Statement Evaluate: \int{xe^{ax}}dx Homework Equations Integration by substitution The Attempt at a Solution I'm on a phone at the moment. My work: http://postimg.org/image/v4hdr5uqx/ The correct answer was: \frac{xe^{ax}}{a}-\frac{e^{ax}}{a^2}

Homework Statement Say for example, a particles velocity was given by the following equation: \vec{V}[SIZE="3"](t) =[SIZE="3"] (2t2-4t3)\hat{i} - (6t +3)\hat{j} + 6\hat{k} If I wanted to find the displacement of the particle between t=1s and t=3s, could I just integrate like this...

$ \displaystyle{ \int { 4 \cos(3x) } \,dx } $ . Hi, for this problem, is this the same as 12*cos*x? Thanks,

Homework Statement ∫x(2^x^3)dx Homework Equations The Attempt at a Solution I've tried using substitution using both x^3 and 2^x^3 as u. I did get pretty far by using log_2 on each side. ∫log_2(x2^x^3)dx=∫(log_2(x)+log_2(2^x^3))dx=∫(log_2(x)+x^3)dx At this point I'm not...

Homework Statement Evaluate: \int \frac{3x}{x^2+2} Homework Equations \int \frac{1}{u} \frac{du}{dx} dx = \ln u + C The Attempt at a Solution I got a horribly wrong answer: \frac{1}{2x}\ln (x^2+2)+C This was done by multiplying \frac{du}{dx} by \frac{3x}{u} This part is what...

I understand \int^{1}_{-1}1-|x|dx = 1 visually just by graphing it and taking the area of the triangle but for the sake of more complicated examples I'm not exactly sure what step I'm messing up when I use the FTOC: |x|= x when x>0, -x when x<0 \int^{0}_{-1}1-|x|dx + \int^{1}_{0}1-|x|dx...

Homework Statement A drag racing car starts from rest at t = 0 and moves along a straight line with velocity given by v = bt^2, where b is a constant. The expression for the distance traveled by this car from its position at t = 0 is: A. bt3 B. bt^3/3 Homework Equations Velocity...

Homework Statement Consider the inner product $$\frac{1}{2\pi}\int_0^{2\pi} \left(\frac{3}{5 - 4\cos(x)}\right) e^{-ikx} dx, \quad k \in \mathbb{Z}, \quad x \in \mathbb{R}.$$ Homework Equations Is there a method to solve this without using the residue theorem, e.g. integration by parts...

Hello Physics Forum, I am trying to find an analytic solution to an equation of the form ∫sin(x/a) exp(b sin(x/a)) dx. I have tried integration by parts and all the usual tricks but can't seem to get anywhere Thanks in advance for your help James

## \frac {1}{4} \int sin^2 (2x)dx = I = \frac {1}{4} [- \frac {1}{2} sin(2x)cos(2x) + \int cos^2 (2x)dx]## when ##u = sin(2x), dv = sin(2x)dx, v= - \frac {cos(2x)}{2}## and ##du = 2cos(2x)dx## Now simplifying ##\int cos^2 (2x)dx## you get ## x - \int sin^2 (2x)dx = x - I## Then, ## I =...

Homework Statement Given \textbf{E}(z,t) = E_{0}cos(kz+ωt)\textbf{i} Find B Homework Equations ∇ x E = -\frac{\partial\textbf{B}}{\partial t}The Attempt at a Solution Taking the curl of \textbf{E} gives (0, -ksin(kz+\omega t), 0) so \frac{\partial\textbf{B}}{\partial t} = (0,ksin(kz+\omega...

Homework Statement ∫(0,1) √x/√[3]1-x Homework Equations \Gammap\Gammaq/\Gammap+q The Attempt at a Solution p-1=1/2 →p=3/2 q-1=-1/3 →q=2/3 β(3/2,2/3)=\Gamma(3/2) \Gamma(2/3)/\Gamma(13/6) \Gamma3/2=1/2\Gamma(1/2)=√π/2 \Gamma2/3=-1/3 \Gamma13/6=7/6 1/6=7/36 β(3/2,2/3)=-6√π/7

Okay so the answer in b) is mgR how is this possible when we integrate ? The work is the external force right? Secondly the F inside the integral is the mg sin(theta) the force of gravity? dr ---> pi R (semicircle)

OK, I'm new to multi-variable calculus and I got this question in my exercises that asks me to integrate e^{-2(x+y)} over a diamond that is centered around the origin: \int\int_D e^{-2x-2y} dA where D=\{ (x,y): |x|+|y| \leq 1 \} I know that the region I'm integrating over is symmetric...

\(r = r(x, t)\), \(q = q(x, t)\), \(\sigma_3 = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix} \) I have the equation \begin{align} \frac{\partial\mathbf{V}_{-1}^{(D)}}{\partial x} &= \frac{i}{2}\begin{bmatrix} -(qr)_t & 0\\ 0 & (qr)_t \end{bmatrix}\\ \mathbf{V}_{-1}^{(D)} &= \alpha\sigma_3 +...

Homework Statement Evaluate the following integral by integrating the corresponding complex function. \int_{-\infty}^\infty \frac{dx}{x(x^2+x+1)} Homework Equations Cauchy's Residue Theorem for simple pole at a:Res(f;a)=\displaystyle\lim_{z\rightarrow a} (z-a)f(z) The Attempt at a...

Mod note: Edited the LaTeX so that the exponents show up correctly.[/color] Homework Statement This is from my Calculus II exam practice papers. We're currently dealing with different substitution methods (whichever apply to the given problem).Homework Equations \int \frac {\sqrt{1 - x^2}}...

This is something that has bothered me for some time, and I can't seem to find any threads on here about it. In a lot of my undergraduate courses in physics, we talk about integrating something physical to infinity. For example, in electrostatics, we talk about the work needed to assemble a...

Homework Statement Show that \int_{-\infty}^{+\infty} \frac{x-1}{x^5-1}dx = \frac{4\pi}{5}sin(\frac{2\pi}{5}) The Attempt at a Solution This is actually a piece of work from a complex analysis module (not sure if it belongs in this part of the forum or in the analysis section) I...

My book loves to represent the delta function as: δ(r-r')=∫-∞∞exp(i(r-r')k)dk Now I can understand this formula if the integration was over the unit circle since. But this is an integration for which the antiderivative as no meaningful limit as x->±∞

Question: Suppose f is continuous, f(0) = 0, f(1) =1, f'(x) > 0, and ∫01f(x)dx = 1/3. Find the value of the integral of f-1(y)dy One solution is to assess the function as if it were a function of y. I understand that method and have arrived at the answer. But I am curious to see if there...

For example, if you have the function f(x) = x2 then find: d/dx any number3x∫ t2dt Why must the dx in d/dx ∫f(t)dt always match the upper limit in order to compute the integral? Why is the lower limit of no concern? I know that you must take chain rule into consideration and change 3x to...

Is there a proof that shows why indefinite integrals cannot be assessed when there are an infinite number of discontinuities but definite integrals are can only be assessed when there is no discontinuities? Why does the fact whether there is one or infinite make a difference? Any mathematic...

First a little warm up problem. Suppose g:\mathbb{R}^N\to\mathbb{C} is some fixed function, and we want to find f:\mathbb{R}^N\to\mathbb{C} such that g(x) = u\cdot\nabla_x f(x) holds, where u\in\mathbb{R}^N is some constant. The problem is not extremely difficult, and after some work...

Homework Statement Compute the standard inner product <f,g> between two one-dimensional functions f(x) = rect(0.5 + x) and g(x) = rect(0.5x), which both depend on the argument x is a member of ℝ. Homework Equations Clearly, we must solve the following : <f,g> = ∫f(x)g(x) dx between...

Homework Statement Integrate ##f(x,y,z) = z## over the region bounded by ##z = 0##, ##x^2 + 4y^2 = 4##, and ##z = x + 2##, Homework Equations None. The Attempt at a Solution I sketched the region in question, but my drawing is so terrible that I'm afraid it'll be little help to anybody who...

I'm trying to integrate the equations of motion for a object. F + mg = ma where F is the drag force, g gravity, a is acceleration, etc... I'm trying to do it numerically and I'm confused about one thing: Since this is a 2nd order vector differential equation, should it be equivalent...

Homework Statement Triple Integral: x^2+y^2+z^2dV over the ball x^2+y^2+z^2 ≤ 9 Homework Equations The Attempt at a Solution so With my integral I had Triple Integral: p^3sin∅dpd∅dθ 0≥p≥3 0≥∅≥∏ 0≥θ≤2∏ Does this look like the correct integral? I swear it is! Yet my answer...

$ kxy \frac{dy}{dx} = y^2 - x^2 \quad , \quad y(1) = 0 $ My professor suggests substituting P in for y^2, such that: $ P = y^2 dP = 2y dy $ I am proceeding with an integrating factor method, but unable to use it to separate the variables, may be coming up with the wrong integrating factor ( x )

Im stuck with this cosh x cos y dx/dy =sinh x sin y after doing I am left with coth x/tan y= dy/dx lost in trying to get y as a function of x due to integrating of trigo

How can I integrate this expression: \[ \int_0^{\infty} \mathcal{J}_1(kR)e^{-kz}dk = \frac{1}{R} \left[1 - \frac{z^2}{\sqrt{R^2 + z^2}} \right] \] where \(\mathcal{J}_1\) is the Bessel function of order 1.

$(1)\;\; \displaystyle \int \cos 2\theta\cdot \ln \left(\frac{\cos \theta +\sin \theta}{\cos \theta -\sin \theta}\right)d\theta$ $(2)\;\; \displaystyle \int \frac{\tan 2\theta}{\sqrt{\sin^6 \theta +\cos ^6 \theta}}d\theta$ I have Tried for (II) :: $\displaystyle \int \frac{\tan...

this may seem simple, but try doing this yourself. I've tried sustituting t=e^x , e^-x. but the problem lies after that. do it and see it for yourself.

Homework Statement Verify, by division, that 2x/(3x+1) = 2/3 - 2/3(3x+1) Hence, evaluate ∫2x/(3x+1) dx I don't understand what to, does the question mean to do long division? Help is much appreciated!

Hi, I don't know if this is the proper part of the forum to ask this, but I'm trying to figure out how I can obtain a certain value from an equation that contains an integral if I want to use experimental data. To keep it simple, the equation looks like this: B= (∫G(f)df)2 / (∫G(f)2df) where...

Homework Statement ∫cos(2x)cos(6x)dxHomework Equations The Attempt at a Solution When I do this one, I seem to get a different answer than my book. The book uses a product to summarize formula, but I hate memorizing formulas and want to do this without it. Here's what I did: Let u=6x then...

Hi there everyone, ∫ sin^2(x-pi/6) dx I have the following integral to solve but am unsure where I should start, I first thought about integrating by parts as I thought you could split it into [Sin(x-pi/6)][Sin(x-pi/6)]. But couldn't seem to figure that out. I was wondering if you could...

Homework Statement The problem that we have been given is to integrate the following: ∫( \frac{4}{2x-1} )dx Homework Equations I understand that the when \frac{a}{ax+b} is integrated, the result is ln(ax+b) + C. The Attempt at a Solution I have been told I need to make the numerator the...

Homework Statement This ( http://www.wolframalpha.com/input/?i=integrate+cos%5E6+(x)+dx+from+0+to+pi%2F2 ) is the integral I am trying to evaluate.: int cos^6 (x) dx from 0 to pi/2 Homework Equations (1 + cos(2x))/2 = cos^2 (x) (1 – cos(2x))/2 = sin^2 (x) sin^2 (x) + cos^2 (x) = 1...

Integral of ... Homework Statement Hi, no directions were given it just says ∫(tan(x/2))^2 dx between 0 and pi. You will get for the integral (1/2 (sin(2x)) - ((1/6)sin(2x))^3 I think that this is OK. Part of the graph of the origonal function dips below the axis so it end up being 0. I...

Homework Statement Hi I am looking at a circle in a Cartesian coordinate system (x, y, z), with center at the point (0, 0, L) and radius R (so the z-axis is normal to the surface of the circle). From the origin (0, 0, 0), I would like to integrate across the circular surface, i.e...

Im doing some complex variable "counter integration" problems and this one came up. I = \oint e ^{\frac{z}{\overline{z}}}dz the integral must be done over a circle with radio r My first attempt was to do it in the exponetial form, so we have this: \frac{z}{\overline{z}} =...

Ok, so I have this differential equation. \[(3x^2y+2xy+y^3)+(x^2+y^2)y\prime=0\] First I needed to check to see if it is exact. \(M=3x^2y+2xy+y^3\) \(N=x^2+y^2\) \(\dfrac{\partial M}{dy}(3x^2y+2xy+y^3)=3x^2+2x+3y^2\) \(\dfrac{\partial N}{dx}(x^2+y^2)=2x+0\) For the integrating factor, I...

Problem: xy'+2y=3x Attempt: Divide by x... y'+\frac{2y}{x}=3 I think I find the integrating factor by doing: e^{\int \frac{2}{x}dx} Not sure if that's right but if it is then the solution to the integral is just 2x. Any help is appreciated

Homework Statement Solve the initial value problem: $$sin(x)y' + ycos(x) = xsin(x), y(2)= \pi/2$$ Homework Equations The Attempt at a Solution Recognizing it as a Linear First-Order Equation:$$\frac{dy}{dx}+y\frac{cosx}{sinx}=x$$ $$P(x)=\frac{cosx}{sinx}$$ Integrating...

$$w'+2w=0\\ \frac { dw }{ dx } =-2w\\ I(x)={ e }^{ 2x }\\ \frac { dw }{ dx } { e }^{ 2x }=-2w{ e }^{ 2x }\\ \int { \frac { dw }{ dx } { e }^{ 2x } } dx=\int { -2w{ e }^{ 2x } } dx$$ Not sure what to do next.

Homework Statement This may seem like a strange question but I need help putting and integral into wolfram alpha/mathematica. I have to find <x>, <x^2>, <p>, and <p^2> for a given wave function. I know the formulas for all these values but the wave function is ψ(x,t)= Axe^[-x^2...

Homework Statement "Show that - \int^1_0 x^k\ln{x}\,dx = \frac{1}{(k+1)^2} ; k > -1. Hint: rewrite as a gamma function. Homework Equations Well, I know that \Gamma \left( x \right) = \int\limits_0^\infty {t^{x - 1} e^{ - t} dt}. The Attempt at a Solution I've tried various substitutions...

Homework Statement y' + (2/t)y = (cost)/(t^2), and the following condition is given: y(pi) = 0Homework Equations The Attempt at a Solution After employing the integrating factor, I find the solution to be: y=e^{-2t} \int e^{2t} \frac{\cos(t)}{t^2} dt. Evidently, this simplifies all the way to...