Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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14 views

Laplace Transform Question Part B and C

Consider the following ODE for y(t): y′′+5y′+6y=e−4t+δ(t−1), y(0)=0, y′(0)=0. (a) Solve this ODE using Laplace transforms. i got this part i believe as i verified it back into the equation for ...
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1answer
21 views

Verify Stoke's Theorem with given information

Verify Stoke's theorem if $\mathbf{v} = z\mathbf{i} + x\mathbf{j} + y\mathbf{k}$ is taken over the hemispherical surface $x^2 + y^2 + z^2 = 1$ , $ z > 0$ Stoke's theorem states the following: ...
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2answers
43 views

Evaluate a certain integral over all space

Evaluate the integral $\iiint e^{-2r} \cos^2\theta \, dV $ over all space. What I have done: I wrote the limit of integration as this: $\int_0^\pi \int_0^{2\pi} \int_0^\infty r^2e^{-2r} ...
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1answer
20 views

Verifying Equivalence with Sec(x) and Identities

I'm trying to prove a couple of different problems and I'm having difficulty proving them on my own and could use a little help and advice. The first thing I needed to prove that this identity is ...
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1answer
28 views

Prove a complex integral function is holomorph and computing the derivative

I need help proving the following statement: Prove $g(z)$ = $\int_{|w-3i|=5} \frac{|w|^2}{(w-z)^4} $ is holomorph over the compliment of the circle {$w||w-3i|=5$} and computing the derivative. Here ...
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4answers
103 views

Verify ∫sec(x) =1/2 ln |(1+sin(x)) / (1-sin(x))| + C

Question says it all, how can I verify the following? $$\int\sec x\ dx=\frac12 \ln \left|\frac{1+\sin x}{1-\sin x}\right| + C$$
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0answers
33 views

Properties of the general solution?

I want find the general solution of the equation: \begin{equation} \frac{1}{2}\left(\frac{\mu_1-\mu_2}{\sigma}\right)^2p^2(1-p)^2 \frac{d^2 u}{dp^2}(p) ...
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1answer
84 views

Find the indefinite integral: $\int { {\sqrt{x+1}} \over {\sqrt{x+2} - \sqrt{x-2}} }dx$

Find the indefinite integral: $$\int { {\sqrt{x+1}} \over {\sqrt{x+2} - \sqrt{x-2}} }dx$$ I don't know how to start, multiplying by ${ {\sqrt{x+2} + \sqrt{x-2}} \over {\sqrt{x+2} + \sqrt{x-2}} }$ ...
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3answers
101 views

Integrate $\int \frac{dx}{x \ln x}\, \mathrm{d}x$ using integration by parts

I know that the integral of $\int \frac{1}{x \ln x}\, \mathrm{d}x$ can easily be obtained through substitution for $u=\ln x$ with the result of $\ln \ln x+C$. My question is if this answer (or an ...
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1answer
20 views

Question about trigonometry substitution integration.

For example, $\int \sqrt{16-9x^2}$. I know that to solve this you've to substitute $x=\frac{4}{3}\sin u$ (Through W.A). But how do I know what to let $x$ be when there is no W.A? Sorry for my english. ...
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1answer
66 views

Use fourier transform to solve second-order differential equation — an “easy” integral?

I have scoured the internet for a fully-explained solution to this problem but have found none: The problem asks to solve this differential equation for $y(t)$ using Fourier Transforms, and then ...
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0answers
31 views

How can I find $\int {\sqrt {{{\left[ 1 - {r k \cos \left( {(w - t )s + p} \right) } \right]}^2} + {{\left( {r w } \right)}^2}}} \mathrm{d}s$?

I have to integrate $$ \int {\sqrt {{{\left[ 1 - {r k \cos \left( {(w - t )s + p} \right) } \right]}^2} + {{\left( {r w } \right)}^2}}} \mathrm{d}s. $$ I have tried solving it with Mathematica, but ...
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2answers
33 views

How to find value of the given integral

Let $\Omega=\{z\in \mathbb C:Im z>0\}$ and let $C$ be the smooth curve lying in $\Omega $ with initial point $-1+2i$ and final point $1+2i$. Find the value of $\displaystyle \int ...
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1answer
25 views

inequality involving norms and integrals

For a square integrable function $f$, is the following true, and if so under what circumstances? \begin{equation} \left\Vert \int_{a}^{b}f\left(t\right)dt\right\Vert _{2}\leq\int_{a}^{b}\left\Vert ...
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3answers
81 views

Find the indefinite integral: $\int {1 \over {x^2 \sqrt {x^2-1}}}dx$ - more simple way?

Find the indefinite integral: $$\int {1 \over {x^2 \sqrt {x^2-1}}}dx$$ I solved it using Integral Substitution where $t=\arccos{1 \over x}$. But is there a more simple way? (not $x = {1 \over ...
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1answer
82 views

Double integral involving incomplete Gamma function

I want to solve an integral of the type: $\int_c^d \frac{1}{x^s} \int_{x}^{\infty} t^{r-1} e^{-t} dt dx = \int_c^d \frac{1}{x^s} \Gamma(r, x) dx$ So, the inner integral is an incomplete Gamma ...
7
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1answer
81 views

Evaluating an integral - is it a two dimensional beta function? This arises from a variant of Goldbach's conjecture.

Let $\gamma>0$. I would like a nice way to prove that $$\int_{\begin{array}{c} 0\leq s,t\leq1\\ s+t\leq1 ...
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1answer
85 views

Upper Riemann sum and area

How do we know that upper Riemann sum is greater than the area under a curve? Isn't it a bit circular? What I mean is that we define area under a curve as a limit of upper and lower Riemann sums ...
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1answer
25 views

In Fourier Series when is it acceptable to just integrate half of period and double the result later to find coefficient?

in finding the coefficient of Fourier Series, $a_0, a_n, b_n$. We integrate the periodic function $f(t)$ over the period $T$. That is $$\frac1T\int^T_0 f(t)\ dt$$ $$\frac2T\int^T_0 f(t)\cos(n\omega ...
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2answers
97 views

How can I convert this tricky complex number into a real number?

The problem statement is: $$∫_0^{\infty}\frac{x^α}{x^3+1}dx$$ for α in the range −1<α<2. $$\huge \frac{2\pi i}{1-e^{\frac{i2\pi (\alpha+1)}{3}}} \frac {e^{\frac{i \pi \alpha}{3}}} { ...
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1answer
36 views

$\int \frac{dx}{a\sin x + b \cos x} $ using complex numbers

I want to do the following integral using complex numbers: $$\int \frac{dx}{a\sin x + b \cos x} $$ Specifically, I plan on using Euler's form : $$e^{ix} = \cos x + i\sin x$$ Then, $$(a\sin x + ...
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1answer
82 views

A comprehensive problem about $\sum {\frac{{n\cosh \left( {nx} \right)}}{{\sinh \left( {n\pi } \right)}}} $

Prolem: ${\mathop {\lim }\limits_{x \to {\pi ^ - }} \mathop {\lim }\limits_{n \to \infty } \left( {{n^2}\left( {1 - \sin \frac{\pi }{{2n}} \cdot \sum\limits_{k = 1}^{n - 1} {\sin \left( {\frac{{\left( ...
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1answer
84 views

How to evaluate integration $\cos(x^2)\ dx$.

I use the substitution x^2=z.With this substitution the integrand becomes cosz/2z^(1/2).Is it possible to integrate w.r.t. z.Please help me.
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72 views

Finding the general solution for this differential equation, where g(t) is an arbitrary continuous function

I have the differential equation $4y''+y=g(t)$, of which I want to find the general solution, if $g(t)$ is an arbitrary continuous function. I also want to find the solution given the initial ...
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0answers
25 views

On the right half-plane, what is an upper bound for $\frac{1}{\log(z+2)}$?

I am trying to estimate some factors in my integrand in complex integration, and I think the upper bound for $\frac {1}{log(z+2)}$ on the semicircle in the right half plane is just $\frac ...
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1answer
47 views

Powers of Sine/Cosine Integral Proof and connection to Fourier Convergence Guidance

Suppose that $f$ is a continuous function on ℝ satisfying $f(x+2\pi) = f(x)$ , If $$\int_{-\pi}^{\pi} f(x) \cos^n(x)\,dx = 0$$ for all $n \ge 0$ and $$\int_{-\pi}^{\pi} f(x) \sin^n(x)\,dx = 0$$ for ...
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1answer
34 views

Calculating volume of revolving function around x then y-axes

Given: $f(x)=(x^2)+1$, $x=1$ & $x=2$ Find volume of enclosed revolved around x-axis $\int_1^2 \pi (x^2 + 1)^2 dx = \pi \int_1^2 (x^2 + 1)^2 dx$ $= \pi \int_1^2 (x^4 +2x^2 +1) dx = ...
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1answer
27 views

How to calculate the integral on surface which cannot be expressed in functions easily?

Surface $S$ is part of $x^2+y^2=1$ between planes $z=0$ and $x+y+z=2$, a vector field $\vec{F}=x\vec{i}+y\vec{j}+z\vec{k}$, what is the value of integration $$ \iint_{S} \vec{F}\cdot\vec{n} dS$$ My ...
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0answers
12 views

What is the Fourier series coefficient of $f(\omega t)$?

Sorry for the unclear title. Normally, to find the coefficient of the Fourier series we do the integral $$a_n=\frac2T\int^T_0 f(t) \cos(n\omega t) \, dt$$ $$b_n=\frac2T\int^T_0 f(t) \sin(n\omega t) \, ...
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28 views

integration over dependent variables

given that $y = f(t)$ , does $$\int t\,dy = ty \text{ ??}$$ I don't think so, as $y$ is not independent of $t$. we can get $$dy = f'(t) \, dt $$ I see it's matter of constants here, but what about ...
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1answer
43 views

Evaluating a logarithmic integral with square roots

We need to evaluate $$I = \int x\ln\left(x^2 + a^2 + \sqrt{x^2 - a^2}\right)\, \mathrm{d}x$$ so we using $u = \ln f(x)$ and $\mathrm{d}v = x$ so that $\mathrm{d}u = \frac{f'(x)}{f(x)}$ and $v = ...
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1answer
51 views

integrate : $\displaystyle\int \frac{x^2}{(3+4x-4x^2)^\frac{3}{2}}dx$

could somebody give me a hint on how to integrate this : $$\displaystyle\int \frac{x^2}{(3+4x-4x^2)^\frac{3}{2}}dx$$ I have the impression that I must do some polynomial division but i have no idea ...
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0answers
29 views

How to understand this table of legendre polynomials?

I don't quite understand what the table is telling me. I understand that h:[-1,1] is the function in the top left hand corner, but that's about all I can figure out. Could somebody please explain ...
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1answer
85 views

Fundamental theorem of calculus applied to a continuous function differentiable except on a countable set

Consider a function continuous $f : [0, \infty) \to \mathbb{R}$ such that $f$ is differentiable except on a set $S$ containing countably many points, and its derivative on $[0, \infty) \setminus S$ is ...
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0answers
14 views

taking derivative under an integral

Suppose I have the following integral $\int_3^{+\infty} x \cdot (x^2 - 8)^{-2} d(x^2)$ Does it equal to the integral $- \frac12 \int_3^{+\infty} ((x^2-8)^{-1})' d(x^2)$ Can I take a derivate of ...
2
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1answer
75 views

Equality in the triangle inequality for integrals

Let's say we have an integrable function $f:\Omega \rightarrow \mathbb{C}$ on some measure space $(\Omega, \Sigma, \mu )$ for which the triangle inequality for integrals holds, i.e.: $$ \lvert ...
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1answer
40 views

Finding Arc Length of a curve

Okay so just now getting into this. It's a rather straight forward topic, but it seems that a lot of questions have a quick and easy way of being solved. I found where that definite integral in the ...
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59 views

How to compute $\int _0^{2\pi }\frac{1-\cos \left(t\right)}{\left(\frac{5}{4}-\cos \left(t\right)\right)^{\frac{3}{2}}}dt$

How to compute $$ \int_{0}^{2\pi}\dfrac{1-\cos(t)}{\biggl(\dfrac{5}{4}-\cos(t)\biggr)^{\dfrac{3}{2}}} dt $$ I'm interested in more ways of computing this integral. My thoughts: I ...
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1answer
20 views

How can I solve this Integral?(complex variables)

$\displaystyle{\int_{C}}{\frac{e^z}{z^3}}dz$, where $C(t)=2e^{2\pi it}+1+i$, and $0\leq t\leq1$ I'm a bit confuse with this integral, the thing is that at the moment I saw it, I though I could use ...
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35 views

Evaluating Hyperbolic Cotangent (coth) Integral

I am working on some simulation, and the paper that I am basing some of the work off of involves several complex integrals. In particular, the one I am trying to solve is $\int_0^\infty ...
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2answers
62 views

Does the “right endpoint rule” yield the same definition of the Riemann integral? [duplicate]

In an analysis class I'm teaching, I stated a different definition of the Riemann integral, which I thought was equivalent to the usual one. But now I'm not so sure. Notation: let $f : [0,1] \to ...
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34 views

Integrals of $2\pi$-periodic functions

Let $f,g,h$ be continuous $2\pi$-periodic functions. In a certain proof the following step is used, and I am having my doubts about it. Is $\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}f(x-t-y)g(t)h(y)dtdy$ ...
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1answer
34 views

Using a transformation to polar coordinates to integrate a function $f : \mathbb R^2 \rightarrow \mathbb R$ over a disk.

If I'm integrating a function $f : \mathbb R^2 \rightarrow \mathbb R$ over a disk $D$ (where $D = \left\{(x,y)\,|\,x^2 + y^2 \le s^2\right\}$), and I want to use the change of variables theorem to ...
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0answers
17 views

Error expansion for trapezoid rule on a multiple integral

Consider a multiple integral over the cube $$If=\int_0^1...\int_0^1 f(x_1,...,x_n)dx_1...dx_m$$ where $f$ is a smooth function. We apply the trapezoid rule $Tf$ on such an integral with step size $h$, ...
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34 views

Error bounds for numerical integration when the function is not smooth

It is well known that error bound for midpoint rule is given by $$ E_M=\int_0^t f(x)dx - h\Sigma_{i=0}^{n-1}f(t_{i+1/2})\leq K \frac{h^2}{24} $$ where $t_{i+1/2}=(i+1/2)h$ and $f''(x)\leq K$. To ...
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4answers
48 views

Struggling to find a Closed Form for an Integral

$$\large{\int_0^\theta (\tan\theta-\sin\theta)\sec^2\theta d\theta}$$ $$$$ The following Integral came up while computing the value of Work performed by a Spring force. I tried to search for the ...
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2answers
53 views

How to integrate exponential * fraction

$$\int_{-\infty}^{\infty}-\frac{1}{2\pi(\omega^2+16)}e^{-i \omega t} d\omega$$ I tried using partial fractions to separate $1/(\omega^2+16)$ into $-1/(8*(4-i\omega))$ and $1/(8*(-4-i\omega))$ but ...
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1answer
20 views

The three dimensional integral

I tried to evaluate such integral: $$I=\int \frac{du_1du_2du_3u_1^{i\eta}u_2^{-i\eta}\delta(1-u_1-u_2-u_3)\theta(u_1)\theta(u_2)\theta(u_3)}{(u_3+au_1u_2)^2} $$ where $\delta(x)$ is a Dirac $\delta$ - ...
1
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1answer
61 views

Integral looks like fraction of elliptic integrals but Mathematica cannot solve it.

I would like to obtain an analytical solution of this integral: \begin{equation} ...
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1answer
41 views

Does this integral have a closed form? $\int^{\infty}_{-\infty} \frac{e^{ikx} dx}{1-\mu e^{-x^2}}$

The original integral contains $\sin [n(x+a)] \sin [l(x+a)]$ but I think this form is simpler: $$\int^{\infty}_{-\infty} \frac{e^{ikx} dx}{1-\mu e^{-x^2}}$$ $\mu <1$, so I can use the Taylor ...