Integrating Definition and 971 Threads

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...

Homework Statement I need to integrate dx/dt=k(a-x)(b-x) and then rearrange to find xThe Attempt at a Solution 1/((a-x)(b-x))dx = k dt Integrating dx using partial fractions: 1/(b-a)×(ln((a-x)/(b-x))=kt+c when t=0 x=0 ∴c=(ln(a/b))/(b-a) then when I rearrange I get...

I have a question regarding what I highlighted in the paint doc. How can they assume that the function is increasing? As a matter of fact its not increasing on the entire interval from 0 to pi.

Hello everyone, What does it mean if I integrate Newton's law of universal gravitation with respect to r. F= GMm/r^2 become 3GMm/r^3 . Is this the work needed to escape a gravitational pull ? Thank you

Hi, I'd like to numerically solve the IVP: x^2 y y''+2x y y'+2 y^2+xy'-x^2 y^3-(x y')^2-y=0,\quad y(x_0)=1,y'(x_0)=0 around the unit circle, x=e^{it}. When I attempt to solve it around the entire circle, I think the integration is veering of course. I believe the solution is single-valued...

Integrating a product of two functions - one "lags" the other I am wondering if there is a way to integrate the following function without first expanding the brackets: \int\limits_{x1}^{x2} x^2\left(x-a\right)^2\,dx The idea behind the question is a bit more complex than I am letting on...

Homework Statement Let X ~ norm(5,10). Find P(X>10).Homework Equations f(x) = \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} F(x) = P(X<x) = \int_{-∞}^x f(u) du The Attempt at a Solution P(X>10) = 1 - P(X<10) P(X<10) = \int_{-∞}^{10} \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} dx =...

Homework Statement A particle of mass m is confined to move in a one-dimensional and Diract delta-function attractive potential V(x)=-\frac{\hbar^2}{m}\alpha\delta(x) where \alpha is positive. Integrate eh time-independent Schrodinger equation between -\epsilon and \epsilon. Let...

Homework Statement Use the method of separation of variables or an integrating factor to find a particular solution of the differential equation that satisfies the given initial condition. y'=x-y+2 ; y(0)=4 2. The attempt at a solution I've used an integrating factor of e^{x} to...

So the problem is ∫(6x+5)/(2x+1)dx. I know the proper way to solve this is to long divide these two expressions and then solve. However, I tried doing it with substitution. u = 2x+1 dx = du/2 I then reasoned that 3u + 2 = 6x+5 since 3(2x+1) + 2 = 6x+3+2 = 6x+5 so I substituted it on top...

Here is the question: Here is a link to the question: Show that the given equation is not exact and find the right integrating factor to make this equation exact? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I'm trying to figure out the general solution to the integral ##\int \frac{d^ny}{dx^n} \, dy##, where n is a positive integer (Meaning no fractional calculus. Keeping things simple.). So far, I have been working with individual cases to see if I can establish a general pattern and then try a...

Homework Statement Integrate -π∫π H(t-π/2)*sin(2t)dt Homework Equations See above. The Attempt at a Solution I can rationalize the slightly simpler integral for the same limits of H(t)*sin(2t) as coming out to 0 due to the definition of the unit step function, but I'm wondering...

Here is the question: Here is a link to the question: Differential equations math question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement \int^{1}_{0}\int^{e^x}_{e^-x}\frac{lny}{y}dydx The attempt at a solution So I am integrating ln(y)/y and I tried it by parts, first with u = ln(y), dv = 1/y, and therefore du = 1/y, and v = ln y but if I use that I get (ln(y))2-\int\frac{lny}{y} again. So I tried...

This is actually related to a post I made earlier in the differential equations forums, but I've since realized that solving the equations themselves is not necessarily the best way to get where I want to go. Perhaps it's better suited to this forum, since it is an integration problem that I...

I've been trying to figure out how to integrate 1/(x+ln(x)) but am not getting anywhere. Mathematica can't do it, and I haven't found it in lists of integrals. Does anyone know if this integral exists in closed form? Same goes for (x+ln(x))/(1+x+ln(x)) Thanks! zeroseven

Homework Statement We define I_{n} = \int_{-∞}^{∞}x^{2n}e^{-bx^{2}}dx, where n is a positive integer. Use integration by parts to derive:I_{n}=\frac{2n-1}{2b}I_{n-1} Homework Equations Parts formula. The Attempt at a Solution So I'm just stuck here, I'm baffled and confused. Firstly if I...

Homework Statement I am trying to integrate the PLanck function to get the Stefan Boltzmann law. After factoring out constants, and substituting x = hv/kT I am left with the following integral: B(T) = ∫ x3/(ex - 1) dx integrated from 0 to ∞ The next step in my notes is that the result...

Homework Statement Hi I have a question about integrating to find the volume of a sphere but before that i need to tell all other results i got or it will not make sence. 1. I needed to show how you get the formula for finding the volume of a sphere by the help of a cirkle with radius r . I...

Homework Statement http://img801.imageshack.us/img801/4227/capturebsw.png Homework Equations The Attempt at a Solution Alright I'm not really sure how to solve this problem. I have redrawn the circuit below without the operational amplifier. In ideal operation amplifiers there's no...

In my diffraction notes, this integral comes up on the page about Babinet's principle: \int ^{y=\infty}_{y=-\infty} \int ^{x=\infty}_{x=-\infty} exp(-i(px+qy)) dx dy = \delta (p,q) I'm not sure how this integral is derived as carrying out the integration and putting in the limits seems to...

This is (perhaps) a tricky question regarding the moment of normal distribution. Let f(x) be the pdf of normal distribution with mean (-σ^2/b) and variance σ^2, where b is just a constant. The goal is to solve the integral ∫ x^3 f(x) dx integrating from 0 to ∞. I am stuck. Any...

I've tried out the two body problem and tried to work out the trajectories with respect to their center of mass frame(located at the origin) as follows (it worked!:smile:): particles = m1 and m2 Fm1 = force of m2 on m1 Fm2 = force of m1 on m2 a = Gm2\hat{r} / r2 \hat{r} is the unit vector...

Using cos^4\theta - sin^4\theta = cos2\theta and cos^4\theta + sin^4\theta = 1 - \frac{1}{2}sin^22\theta evaluate: (i) \displaystyle \int_0^{\frac{\pi}{2}} cos^4\theta \ d\theta adding the two identities given I get: 2cos^4\theta = cos2\theta + 1 - \frac{1}{4} + \frac{1}{4} cos4\theta...

Homework Statement Apologies for the vague title, I'm not reall sure what I'm look at here! I am doing some revision on solving the Friedmann equations, and in a lot of cases I end up having to integrate a function that looks like (xan - 1)-1/2 da = dt where a is a function of t, x is a...

Homework Statement Find the integral of This was the question. There is a way to do it by long division but I am confused with Long division. Instead I tried to do by the method below but I failed... Homework Equations NoneThe Attempt at a Solution I thought maybe I could reduce the...

Homework Statement ∫arctan(√x)dx. Using the substitution √x=t: ∫arctan(√x)dx = ∫arctan(t)dt2 This is what I've got written in a solution manual. I don't see why the dt would be squared. Could anyone care explaining me? thanks

Homework Statement Consider the general linear homogeneous second order equation: P(x)y'' + Q(x)y' + R(x)y = 0 (1) We seek an integrating factor μ(x) such that, upon multiplying Eq. (1) by μ(x), we can write the resulting equation in the form [μ(x)P(x)y']' + μ(x)R(x)y = 0...

Homework Statement \Psi(x,t) = \int^{\infty}_{-\infty} C(p)\Psi_{p}(x,t) dp is a solution to the Schroedinger equation for a free particle, where \Psi_{p}(x,t) = Ae^{i(px-Ept)/\hbar}. For the case C(p) = e^{-(p-p_{0})^{2}/\sigma} where \sigma is a real constant, compute the wavefunction...

Homework Statement This is an arc length problem in three dimensions. I was given the vector r(t)=<et, 1, t> from t=0 to t=1 Homework Equations Arc Length= \int |\sqrt{r'(t)}| dt from t1 to t2 where |\sqrt{r'(t)}| is the magnitude of the derivative of the vector The Attempt at a...

Homework Statement Find the length of the curve between x=0 and x=1. Note: can this be done without a calculator?Homework Equations y = sqrt(4-x^2) The Attempt at a Solution x=2sin∅ dx = 2cos∅ d∅ sqrt(4-4(sin∅)^2) ---> 2cos∅integral (0 to 1) sqrt(1+(-2sin∅)^2) integral (0 to 1)...

Hello, I'm having problem understanding the output of the integrating op amp. https://www.dropbox.com/s/djpg0uhi465ot86/download2.png https://www.dropbox.com/s/djpg0uhi465ot86/download2.png https://www.dropbox.com/s/p9bp9tlyykhjtvf/download.png...

Am I integrating this right: (x^2 + 3x + 11)/(x+2)^4 ? 1. ∫(x2 + 3x + 4)/(x+2)4dx 2. With these sorts of problems, I think about integration by partial fractions. So in this case, the denominator factors are all the same, so I have to make each one with different exponents. I wrote...

when using the reimann integral over infinite sums, when is it justifiable to interchange the integral and the sum? \int\displaystyle\sum_{i=1}^{\infty} f_i(x)dx=\displaystyle\sum_{i=1}^{\infty} \int f_i(x)dx thanks ahead for the help!

Homework Statement for the following integrals, am I allowed to break them up like so: 1. ∫(1)/(sqrt(16-9x²)³) dx = ∫(1)/(√16)³ · ∫(1)/(√-9x²)³ dx 2. ∫(x²)/(sqrt(x²-9)) dx = ∫(x²)/(√x²) · ∫(x²)/(√-9) dx 3. ∫(1)/(x²(sqrt(a²+x²))) dx = ∫(1)/(x²) · ∫(1)/(√a²) · ∫(1)/(√x²) dx...

Homework Statement Volume of the region bounded by 1/x^4, y = 0, x = 2, and x = 6 about the axis y = -4 Homework Equations The Attempt at a Solution The height for any shell is x(y) - 2, or \sqrt[4]{\frac{1}{y}} - 2 The radius of any shell is y + 4 So the circumference...

Homework Statement (3x^2y+2xy+y^3)dx+(x^2+y^2)dy=0

given a function f(x) that is piecewise smooth on interval -L<x<L except at N-1 points, is \int_{-L}^L f'(x)dx legal or would i have to \sum_{i=1}^N \int_{x_i}^{x_{i+1}} f'(x)dx‎‎ where x_{N+1}=L and x_{1}=-L also, am i correct that if f(x) is piecewise smooth, then f'(x) is piecewise...

Homework Statement Use parametrisation first, derive the equation including y and p = \frac{dy}{dx} and use the integrating factor method to reduce it to an exact equation. Leave the solution in implicit parametric form. (y')^{3} + y^{2} = xyy'The Attempt at a Solution I'm really lost at...

Here is the question: Here is a link to the question: Integrating Factor Help Please!? - Yahoo! Answers I have posted a link there so the OP can find my response.

Homework Statement Integrate the following: Homework Equations ∫(x^2+x-7)/(x+3) The Attempt at a Solution The only way I can think of solving this would be to split up each term into a separate fraction.

Homework Statement integrate the following:Homework Equations ∫(x/(x-1)^3The Attempt at a Solution i've tried u-substitution, finding an inverse trig function that matched the formula, and still can't figure out how to solve this problem. u-subtitution for u=x gives the same problem...

I am sure that this can be done, but I haven't been able to figure it out, Is there a way to integrate a differential form on a manifold without using the parametric equations of the manifold? So that you can just use the manifold's charts instead of parametric equations? If you a function...

Homework Statement I am trying to integrate the function \int _{-\infty }^{\infty }(t-1)\delta\left[\frac{2}{3}t-\frac{3}{2}\right]dt Homework Equations The Attempt at a Solution I think the answer should be \frac{5}{4} because \frac{2}{3}t-\frac{3}{2}=0 when t=9/4. then (9/4-1) = 5/4...

I'm trying to integrate acceleration from an acceleraometer to find a distance travelled. I have heard all the stories about this not being accurate but i didn't come up with the method I'm just trying to implement an algorithm to do it. I'ts justan estimate for wear rates, not positioning...

Hello- I have a diffusion profile, in which I plot the decrease in concentration versus distance of my sample. I am trying to find the increase in mass of the overall sample. How would I do this? I have integrated the function and am wondering what type of information that would supply...

The integral of 1/x is ln(x). Where does that come from? That always puzzled me. We can continue to take derivatives through x^0 and into the negative integers, and just use the plain old power rule to get the answers. We can do the same for the integral of x all the way from negative...

Hi all, I'm trying to integrate the function below with respect to x exp(ix)-exp(-ix) With infinity and negative infinity as the limits. Would the integration be possible?