Integral Definition and 1000 Threads

I was trying to solve a problem involving work , as we know : w = \int_{a}^{b} \vec{f}.d\vec{s} but in my problem the path was cyrcular , so how to evaluate this kind of integral ?

Homework Statement Is the solution provided by the author wrong ? Stokes theorem is used to calculate the line integral of vector filed , am i right ? Homework EquationsThe Attempt at a Solution To find the surface integral of many different planes in a solid , we need to use Gauss theorem ...

$\textsf{evaluate}$ \begin{align} \displaystyle {I}&={\int{\frac{x+2}{x^2+1}dx}}\\ &=\int{\frac{x}{x^{2}{+1}}dx{\ +\ 2}\int{\frac{1}{x^{2}{+1}}}}{\ }dx\\ u&=x^{2}+1 \therefore \frac{1}{2x}du=dx\\ x&=\sqrt{u-1}\\ \end{align} ...? $\textit{calculator answer.?}$ $\dfrac{\ln\left(x^2+1\right)}{2}...

I am trying to find primitives to the rational function below but my answer differs from the answer in the book only slightly and now, I am asking for your help to find the error in my solution. This solution is long since I try to include all the steps in the process. The problem $$ \int...

I have been working on this for several days but getting nowhere. Any help would be great. \begin{align} &\int_0^x dy\,y^2 \cos(y^2) C^2 \!\!\left(\!\frac{\sqrt{2}\,y}{\sqrt\pi}\!\right) \end{align} In reality only the first one is causing me troubles, however I have pasted the entire...

Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...

I'm reading the path integral chapter of Schwartz's "Quantum Field theory and the Standard model". Something seems wrong! He starts by putting complete sets of states(field eigenstates) in between the vacuum to vacuum amplitude: ## \displaystyle \langle 0;t_f|0;t_i \rangle=\int D\Phi_1(x)\dots...

Homework Statement Find the green's function for y'' +4y' +3y = 0 with y(0)=y'(0)=0 and use it to solve y'' +4y' +3' =e^-2x Homework Equations ##y = \int_a^b G*f(z)dz## The Attempt at a Solution ##\lambda^2 + 4\lambda + 3 = 0 \to \lambda = -1,-3## ##G(x,z) = \left\{ \begin{array}{ll} Ae^{-x}...

Homework Statement Find the green's function for y'' +2y' +2y = 0 with boundary conditions y(0)=y'(0)=0 and use it to solve y'' + 2y' +2y = e^(-2x) Homework Equations ##y = \int_a^b G(x,z)f(z)dz## The Attempt at a Solution I'm going to rush through the first bit. If you need a specific step...

Hello, I am having trouble with solving the problem below The problem Find all primitive functions to ## f(x) = \frac{1}{\sqrt{a+x^2}} ##. (Translated to English) The attempt I am starting with substituting ## t= \sqrt{a+x^2} \Rightarrow x = \sqrt{t^2 - a} ## in $$ \int \frac{1}{\sqrt{a+x^2}}...

Consider the integral ##\displaystyle \int_{-\infty}^\infty \frac{e^{-|x|}}{1+x^2}dx ##. I should be able to use contour integration to solve it because it vanishes faster than ## \frac 1 x ## in the limit ## x \to \infty ## in the upper half plane. It has two poles at i and -i. If I use a...

Dear "General Math" Community, my goal is to calculate the following integral $$\mathcal{I} = \int_{-\infty }^{+\infty }\frac{f\left ( \mathbf{\vec{x}} \right )}{\left | \mathbf{\vec{c}}- \mathbf{\vec{x}} \right |}d^{3}x $$ in the particular case in which f\left ( \mathbf{\vec{x}} \right...

Homework Statement Calculate ##\iint { y+{ z }^{ 2 }ds } ## where the surface is the upper part of a hemisphere with radius a centered at the origin with ##x\ge 0## Homework Equations Parameterizations: ##\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le...

Homework Statement I have calculate my double integral using wolfram alpha , but i get the ans = 312.5 , but according to the book , the ans is = 0 , which part of my working is wrong Homework EquationsThe Attempt at a Solution Or is it z =0 , ? i have tried z = 0 , but still didnt get the...

Homework Statement Homework Equations k∫[ƒ(x)]n ƒ'(x) dx The Attempt at a Solution i tried to using algebraic substitution to determine that i had let u = 1-x or X2-2x+1 or x or root(x) but it still cannot solve it. Please give me hint how to solve it. Thank you [/B]

I have: $$\int_{1}^{3} \frac{1}{\sqrt{3 - x}} \,dx$$ I can do $u = \sqrt{3 -x}$, and $du = \frac{1}{2 \sqrt{3 - x}} dx $ and $dx = 2 \sqrt{3 - x} du $. Plug into original equation: $$\int_{1}^{3} \frac{2 u }{u} \,du$$ and $2 \int_{1}^{3} \,du = 2u = 2 \sqrt{3 - x} + C$ So $(2\sqrt{3 - 3})...

I have: $$\int_{1}^{2} \frac{1}{x lnx} \,dx$$ I can set $u = lnx$, therefore $du = \frac{1}{x} dx$ and $xdu = dx$. Plug that into the original equation: $$\int_{1}^{2} \frac{x}{x u} \,du$$ Or $$\int_{1}^{2} \frac{1}{ u} \,du$$ Therefore: $ln |u | + C$ and $ln |lnx | + C$ So I need to...

From the path integral approach, we know that ## \displaystyle \langle x,t|x_i,0\rangle \propto \int_{\xi(0)=x_i}^{\xi(t_f)=x} D\xi(t) \ e^{iS[\xi]}##. Now, using ## |x,t\rangle=e^{-iHt}|x,0\rangle ##, ## |y\rangle\equiv |y,0\rangle ## and ## \sum_b |\phi_b\rangle\langle \phi_b|=1 ## where ## \{...

I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...

Homework Statement Let ##f(x,y)=(xy,y)## and ##\gamma:[0,2\pi]\rightarrowℝ^2##,##\gamma(t)=(r\cos(t),r\sin(t))##, ##r>0##. Calculate ##\int_\gamma{f{\cdot}d\gamma}##. Homework EquationsThe Attempt at a Solution The answer is 0. Here's my work. However, this method requires that you are...

Homework Statement I am having question with part c , for both c1 and c2 , here's my working for c1 , i didnt get the ans though . My ans is -5 , but the given ans for c1 and c2 is 27 , is the ans wrong ? Or which part i did wrongly ? Homework EquationsThe Attempt at a Solution

Hello. I have a difficulty to understand the branch cut introduced to solve this integral. \int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]} here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...

$\int_{3}^{\infty} \frac{1}{\sqrt{x} - 1} \,dx$ I need to find if this converges or diverges. I'm trying u-substitution, so $u = \sqrt{x} - 1$. Therefore, $du = \frac{1}{2\sqrt{x}} dx$. I'm not sure how to proceed from here.

Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...

I assume this is a simple summation of the normal components of the vector fields at the given points multiplied by dA which in this case would be 1/4. This is not being accepted as the correct answer. Not sure where I am going wrong. My textbook doesn't discuss estimating surface integrals...

Evaluation of $\displaystyle \int^{\frac{1}{2}}_{0}\frac{1}{(1-2x^2)\sqrt{1-x}}dx$ $\bf{Try::}$ Let $\displaystyle I = \int^{\frac{1}{2}}_{0}\frac{1}{(1-2x^2)\sqrt{1-x}}dx$ Put $1-x=t^2\;,$ Then $dx=-2tdt$ So $\displaystyle I = \int^{1}_{\frac{1}{2}}\frac{2t}{\left[1-2(1-t^2)^2\right]t}dt =...

Please anyone can help solve this integral equation e^t+e^t ∫ (t, 0 ) e^(-τ) x f(τ) dτ

I know integral of (2x+1)/(x^2+x). but i don't know integral of (x^2+x)/(2x+1). I'm very curious... please answer me...

Sorry where is pi/4 coming from in the line integral(section 3)? because i think it should be 1/2=tan(theta) which theta is 26.5651... it is impossible that the angle is pi/4? where is pi/4 coming from inside the circle? thank

Homework Statement Sorry that I am not up on latex yet, but will describe the problem the best I can. On the interval of a=1 to b= 4 for X. ∫√5/√x. Homework EquationsThe Attempt at a Solution My text indicates the answer is 2√5. I have taken my anti derivative and plugged in b and subtracted...

Homework Statement No actual work, could just use some assistance in understanding formulas involving the centroid of an object, specifically with integrals. For example, how would you understand the following formula(s) (as seen in part 2)? I understand that the centroid is the sum of all the...

I have $$\int_{}^{} \frac{1}{\sqrt{1 - x^2}} \,dx$$ I can let $x = \sin\left({\theta}\right)$ then $dx = cos(\theta) d\theta$ then: $$\int_{}^{} \frac{cos(\theta) d\theta}{\sqrt{1 - (\sin\left({\theta}\right))^2}}$$ Using the trig identity $1 - sin^2\theta = cos^2\theta$, I can simplify...

Homework Statement Transform given integral in Cartesian coordinates to one in polar coordinates and evaluate polar integral. ##\int_{0}^3 \int_{0}^x \frac {dydx}{\sqrt(x^2+y^2)}## Homework EquationsThe Attempt at a Solution I drew out the region in the ##xy## plane and I know that ##0 \leq...

Hello I have tried gaussian integrals does gaussian integrals have this general form formula? if not then weather i do integration by parts or what just needed a hint to solve it correctly

Homework Statement I'm having some trouble evaluating the integral $$\int^\infty_{-\infty} \frac{\sqrt{2a}}{\sqrt{\pi}}e^{-2ax^2}e^{-ikx}dx$$ Where a and k are positive constants Homework Equations I've been given the following integral results which may be of help $$\int^\infty_{-\infty}...

If a brick is pulled across the floor by a rope thruogh a pulley, 1 meter above the ground - and work = W, where W = 10N , (in Newton).Show that the horizontal component of W, which is pulling the brick has the size \frac{10x}{\sqrt{1+x^2}} (*) Use this to calculate the amount of work needed...

Does there exist and analytical expression for the following integral? I\left(s,m_{1},m_{2},L\right)=\sum_{\boldsymbol{n}\in\mathbb{N}^{3}\backslash\left\{ \boldsymbol{0}\right\}...

Homework Statement A 12 foot light pole stands at the corner of an 8 foot by 10 foot rectangular picnic blanket spread out on the ground. A bee flies in a straight line from a point P on the pole to a point Q on the blanket. Set up a multiple integral whose value represents the average length...

This is my first time using this site so please excuse me if I missed any guidelines. 1. Homework Statement A plastic rod having a uniformly distributed charge Q=-25.6pC has been bent into a circular arc of radius R=3.71cm and central angle ∅=120°. With V=0 at infinity, what is the electric...

I am trying to evaluate the following integral. ##\displaystyle{\int_{-\infty}^{\infty}f(x,y)\ \exp(-(x^{2}+y^{2})/2\alpha)}\ dx\ dy=1## How do you do the integral above?

Homework Statement $$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##. Homework EquationsThe Attempt at a Solution Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have...

So, ∫x/(1-x)... can I solve this as a power series ∫(x*Σ x^n) = ∫(Σ x^(n+1))= (1/(n+2)*Σ x^(n+2))? Is this correct? I know there is other ways to do it... But should this be correct on a test? This solution is more fun..

Is it possible to do an integral of f(x)*x without knowing f(x)?

Homework Statement Determine ##\int_{0}^{2}\sqrt{x}dx## using left riemann sums Homework Equations ##\int_{a}^{b}f(x)dx = \lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}(\frac{b-a}{n})f(x_i)## The Attempt at a Solution [/B] ##\frac{b-a}{n}=\frac{2-0}{n}=\frac{2}{n}## ##\int_{0}^{2}\sqrt{x}dx =...

Hi, what kind if integration used on equation 1 so it turned into equation 2? this does not look like integration by parts. and where (x-x0) appeared from instead of (k-k0) ? thanks for your help.

Consider the definite integral ∫202x(4−x2)1/5 dx. What is the substitution to use? u= 4-x^2 Preview Change entry mode (There can be more than one valid substitution; give the one that is the most efficient.) For this correct choice, du/dx= -2x Preview Change entry mode If we make this...

[Question] So I was thinking about Physics for some time and for the sake of curiosity I've came with this problem: Let's say we have a liquid flowing into a system with infinite space. The flow is constant ( F ) The liquid decays over time with a half life ( λ ) We're looking for the Total...

Homework Statement Evaluate the following integral: ∫0∞ √(x)* e-x dx Homework Equations ∫0∞ e-x2 dx = (√π)/2 The Attempt at a Solution So far this is what I've done: u = x1/2 du = 1/2 x-1/2 2 ∫ e-u2 u2 du Now, I'm not really sure what to do? Or if what I've done so far is leading me down...

I'm going through the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman. On Page 138 (below) ) The authors take the derivative of a definite integral and end up with a definite integral plus another term. How did...

Is this question correct? We are given to evaluate: \int_0^2 \left(e^x-e^{-x}\right)^2\,dx 2\left(\frac{1}{2}\sinh(x)-x\right) 2\left(\frac{1}{2}\sinh(2\cdot2)-2\right)-2\left(\frac{1}{2}\sinh(2\cdot0)-0\right) \sinh(4)-4