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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1answer
55 views

Integration of an equation containing Legendre Polynomials

Consider the Integration $\int_{-1}^{1}x^{2}P_{n+1}(x)P_{n-1}(x)dx$ where, $P_{n+1}(x) ,P_{n-1}(x)$ are Legendre Polynomials Applying integration by parts,we get $x^{2}\int_{-1}^{+1}P_{n+1}(x)P_{n-...
1
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1answer
20 views

Integral with more-dimensional substitution variables

Good day, In the lecture of partial differential equations we had the following transformation: $$\int_{||\nu||=1} h(x+\nu c t, \tau) d\nu = \frac{1}{c^2 t^2} \int_{||y-x||=ct} h(y,\tau) dy$$ for $...
1
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1answer
71 views

How to integrate this looking simple ODE?

I meet an ODE about $V(\theta)$ $$\frac{d^2V}{d\theta^2}+\frac{1}{2V}=0.$$ But I can not figure out how to integrate it to yield $$\left(\frac{dV}{d\theta}\right)^2+logV=C_1$$ or $$\theta=\int^V\...
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1answer
50 views

Calculus 2 -Calculate the following Integral

Calculate $\int_1^2 f(x) dx $ Assuming $ \int_0^1 f(x) dx = -89$ $\int_0^2 f(x) dx = 87 $ $\int_1^4 = -21 $ Things i've tried 1) $\int_1^2 f(x) dx = \int_1^4 f(x)dx- \int_0^1 f(x) dx - \int_0^...
10
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1answer
138 views

Integral $\int_0^1(x(1-x))^n\frac{d^n}{d^n x}(\log x \cdot\log (1-x))dx$

While playing around with the first values of the integral $$ I_n:=-\int_0^1\left(x(1-x)\right)^n\frac{d^n}{d^nx}\left(\log x \cdot\log (1-x)\right){\rm d}x, \quad \quad n=1,2,3,\cdots, $$ I ...
2
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0answers
100 views

Help on solving integral equation

Hello Stack Exchange Users, I was working on an integration problem, and I "simplified" the integral to the following: $$\int \limits_0^{2\pi} (7.625+.275 \cos(4x))^{1.5} \cdot (A \cos(Nx) + B \sin(...
2
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0answers
34 views

Characteristic function of $\chi^2$ distribution with $n$ degrees of freedom

I'm computing the formula for the characteristic function of the random variable $X \sim \chi^2(n), $ $n\in\mathbb{N}$. After some substitutions in the integral and some messing around with certain ...
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0answers
45 views

Bound on the following integral using integration by parts

Let $W: \mathbb{R}^s \rightarrow [0,1]$ be a smooth function supported on $[0,1]^s$ that satisfies $$ \left| \frac{\partial^k }{\partial x_{i_1} \cdots \partial x_{i_k}} W(\mathbf{x}) \right| \leq \...
2
votes
3answers
152 views

Integrating $\int \frac{u \,du}{(a^2+u^2)^{3/2}}$

How does one integrate $$\int \frac{u \,du}{(a^2+u^2)^{3/2}} ?$$ Looking at it, the substitution rule seems like method of choice. What is the strategy here for choosing a substitution?
2
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2answers
63 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: $$e^{-inx}\...
2
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0answers
83 views

A limit using integral to compute

Is it possible to compute the following limit by using integral? $$\lim_{n‎ ‎\rightarrow ‎\infty} ‎‎\left( ‎‎\dfrac{1}{n^2+1}+‎\dfrac{2}{n^2+2}+‎\cdots ‎+‎\dfrac{n}{n^2+n}‎‎‎\right)$$
1
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1answer
46 views

Proof that $ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4 $?

Can one prove that $$ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3),\text{ for }2<\Re(d)<4? $$ I would prefer using the methods of contour integration.
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1answer
146 views

Infinite intersection of monotonic Jordan measurable sets is also measurable

I want to show that if $E_1 \supset E_2 \supset....$ is a monotonic sequence of Jordan measurable sets, so $$Z = \bigcap_{k=1}^\infty E_k$$ is of Lebesgue measure null, then Z is also Jordan ...
3
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1answer
101 views

How to calculate $\int_{-\infty}^{\infty}\frac{x^2}{\cosh(x)}\mathrm{d}x$ [duplicate]

I know the poles are $z=i\pi/2+i n\pi$ and therefor I got an rectangular contour for the integration which wasn't so useful. I also know with change of variables I can get to $\int_{0}^{\infty}\frac{\...
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0answers
19 views

Integral Inequality and Stochastic Dominance

I need to prove that if $x\leq 0$ and $\alpha \in (1,2]$, then $$\frac{1}{\pi}\int_{-\infty}^{x}(x-z) \left( \int_{0}^{\infty}\cos(tz)\cdot t^{\alpha}\cdot\ln(t)\exp\left(-t^{\alpha} \right)dt \right) ...
0
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1answer
47 views

Asymptotic expansion question

How may I use Watson's Lemma to find the full asymptotic expansion for; $$ I(\lambda)= \int_0^\infty e^{-\lambda(1+s)}ln(1+s^2)ds $$ as $\lambda \rightarrow \infty$. Thanks in advance
1
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1answer
31 views

How to prove that: $\int_{0}^{\infty} \frac {1}{\frac 1 2(e^x-e^{-x} )\cdot x} dx$ Diverges.

How to prove that: $\int_{0}^{\infty} $$ \frac{1}{\sinh(x)\cdot x }dx=\int_{0}^{\infty} \frac {1}{\frac 1 2(e^x-e^{-x})\cdot x} dx$ Diverges. SOLUTION ATTEMPT: I thought about separating this ...
3
votes
2answers
389 views

Evaluate this integral without knowing the function

I'm an high-speed aerodynamics student but have some problems with some math expressions. We know that: $$\int_0^c\dfrac{\partial z_e(x)}{\partial x}dx=z_e(c)-z_e(0)$$ I'm having trouble to do such ...
3
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3answers
112 views

How to show that $\int_0^{\infty} dx \frac{\log{x}}{1+x^2}$ is zero using complex analysis

I want to show this using contour integration, the appropriate contour is a keyhole I think.
2
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3answers
190 views

Evaluating $\int_{-\pi}^{\pi} (e^{ix} + e^{-ix})^n dx $

In an exercise following identity is used: $$ \int_{-\pi}^{\pi} (e^{ix} + e^{-ix})^n dx = \begin{cases} 0, \hspace{2.1cm} n = 2m+1 \\ 2\pi {2m \choose m}, \hspace{1cm} n=2m. \end{cases}, $$ Does ...
2
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0answers
48 views

Integrability of $f\left(\frac1n\right)=\frac1n$ and else $f(x)=-1$

Let $f:[0,1]\to\mathbb{R}$ be defined as $f(x)=\cos(x)$ if $x=\frac1n$ for a natural number $n\ge1$ and $f(x)=-1$ else. Does $$ \int_0^1f $$ exist? I think it is because the problematic region around ...
0
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1answer
65 views

Volume bounded between an Ellipsoid and a Cone?

I'm a bit confused about how I would be able to find the volume bounded by a cone of known theta and an oblate spheroid of b = c. I'm trying to use triple integrals for the solution, and I think I ...
0
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1answer
36 views

Calculate the limit $ \lim_{x\to\infty} \int_{0}^{1} g(xz) dz$.

let $g(x)$ be a continuous function s.t. for each $x \ge 0$, $\lim_{x\to\infty} g(x)=L \ne 0$. calculate the limit : $$ \lim_{x\to\infty} \int_{0}^{1} g(xz) dz$$ SOLUTION ATTEMPT: I'm thinking ...
2
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0answers
58 views

The function integrated over the two-sphere is smooth in the radial parameter

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a smooth function. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ $$ h(w) = \int_{S^2} d\Omega(\hat{n})\,f(\sqrt{|w|}\, \hat{n}) $$ where $d\...
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0answers
36 views

The Deconvolution Integral

The standard 1D continuous convolution integral is defined as: $$y(t) = h(t)*x(t) = \int^{+\infty}_{-\infty}h(\tau)\cdot x(t-\tau)\ d\tau$$ Using fourier transform, $$Y(j\omega) = X(j\omega)\cdot H(...
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2answers
55 views

How to integrate something already having integration.

The textbook wrote, \begin{align} g_n(t)&=\int_0^tf(t-s)g_{n-1}(s)ds, &n=2, 3, \cdots, &&0 \le t \le \infty \\ G_n(t)&=\int_0^tF(t-s)dG_{n-1}(s), &n=2, 3, \cdots, &&0 \...
1
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1answer
71 views

How to integrate $f'(\ln x)/x$ given a table of values of $f$ and $f'$?

My professor gave me this question without any description. $$\int_1^e \frac{f'(\ln x)}{x}\,dx$$ where $f$ has the following data: \begin{array}{r|lllll} x& 0 & 1 &\frac{\pi}{2}...
2
votes
2answers
175 views

how to calculate $\int_{0}^{\infty}\frac{x}{\sqrt{e^x-1}}\mathrm{d}x$

I was trying to solve another integral when then I reached this, I've no idea of how to select the contour for the integration.
0
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0answers
20 views

A double integral with parameters

Please help me solve this integral: $$\frac{1}{\pi}\int\int_{-tz \in \left[ 0,\frac{\pi}{2} \right]} \cos(tz)\left[ (x-z)\left( e^{-t^{\alpha_1}}-e^{-t^{\alpha_2}} \right) \right] \, dt \, dz$$ ...
3
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4answers
76 views

If $\int_{-x}^{x}f(t)dt=x^3-x^2+x+1$, then find $f(-2)+f(2).$

If $$\int_{-x}^{x}f(t)dt=x^3-x^2+x+1,$$ then how can I find $f(-2)+f(2)?$ I tried to use the derivative of integral but I get $f(2)-f(-2)=9.$
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1answer
46 views

Evaluating $\int \frac{2\ dx}{\sin x \cos x}.$ [duplicate]

Evaluate the following $$\int \frac{2\ dx}{\sin x \cos x}.$$
1
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2answers
34 views

integral of a square root function by substitution.

A practice problem: $$\int \sqrt{x^2+9}\ dx $$ So what I did was to substitute $x$ with $\tan \theta$, which yields $$\int \sqrt{9\tan^2\theta+9}\ dx $$ Then I brought the 9 out $$\int ...
2
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0answers
20 views

Riesz potential of a set and its complement

Let $F\subset [0,1]$ be a closed set, $G = [0,1]\setminus F$, $\alpha \in(1,2)$. Is there a simple condition on $F$ under which the integral $$ \int_F\int_G \frac{dx\,dy}{|x-y|^{\alpha}} $$ is finite? ...
0
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2answers
56 views

Reverse Power Rule integration.

Ok, so I am confused about the following; When we have a polynomial, say $P(x)$, and we want to solve an integral where $P(x)$ is raised to a certain power, for example; $$\int (P(x))^adx$$Why can we ...
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2answers
114 views

Can we interchange the limit and integration for $\int f_n(x)\,\mathrm{d}x$? [duplicate]

In connection with this question about computing integrals of the form $$\mathcal{I}_n=\int_0^\infty f_n(x)\,\mathrm{d}x=\int_0^\infty \frac{\mathrm{d}x}{e^x+x^n}$$ I noticed an interesting trend ...
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0answers
20 views

Integral with Simpson's method not converging

I'm trying to use Simpson's rule to integrate the following function in a program: $$\int_{z_a}^{z_b}\frac{Cf(z)}{(C^2 - f(z)^2)^{3/2}}\,dz$$ where $C$ is a constant and $f(z)$ are interpolated ...
2
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0answers
67 views

Integral of a Gaussian with Trigonometric functions Involved

I am having a difficult time evaluating an integral unlike any integral I have seen before. To get right into things here is the integral: $$\frac{A}{\sigma_o\sqrt{2\pi}}\int_{-\infty}^\infty [\sin(...
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0answers
57 views

Is it possible to perform Integration in this equation?

I have been working on a problem for a long time and have finally arrived at this differential equation. The problem is simple, which surfaces obey the Reflection Property. Now there are several ...
1
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3answers
81 views

Find $\int x\ln(x^2e^{x^2})\,dx.$

How can I find the following integral $$\int x\ln(x^2e^{x^2})\,dx.$$ Which substitution may I use to solve the integral?
1
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1answer
22 views

Chasle relation with arbitrary ordering of endpoints.

Consider the Chasle relation : $$\int_a^c f(x) dx= \int_a^b f(x) dx+ \int_b^c f(x) dx\,$$ Is this only true for $a\leq b\leq c$ or is it true for any ordering of $a,b$ and $c$ ? In the case $a\leq b\...
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0answers
33 views

2-Dimensions Integral convergence

Does the following integral converge on $\mathbb{R}^2$: $\int \int \frac{log(x^2+y^2)}{x^2+y^2}dxdy$ I found that inside the unit circle the integral is -$\infty$ and outside the unit circle its +$\...
3
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2answers
33 views

$\sum_{k=1}^{n} \frac{1}{k} \geq \int_1^{n+1} \frac{1}{x} dx$ Inequality

$$\sum_{k=1}^{n} \frac{1}{k} \geq \int_1^{n+1} \frac{1}{x} dx$$ I don't see how you reach this inequality, or rather why it is correct. The context of this problem was the following: Show that $$...
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votes
3answers
68 views

Integrate $\int \frac{x+3}{2x^2+x+3}\ dx$ [closed]

$$\int \frac{x+3}{2x^2+x+3}\ dx$$ How should I approach this?
0
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1answer
90 views

Double integration problem, how to integrate $e^{x^2}$?

What is the value of $$\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^ye^{-\frac{1}{2}\left(x^2+y^2\right)}dxdy$$ I draw the region of integration tried to change the order but still i don't know ...
0
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0answers
41 views

Is their a special way to factor when doing integrals

(Very sorry this could probably be cleared up if I had rights to comment, however I am new here) Anyways, I was going on stack overflow looking for help on a problem I keep getting wrong. For the ...
0
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0answers
30 views

Is this integration correct?

I am checking some papers. $$ \int_{0}^{\frac{D_{ab}}{2k^2}} \left( v_{ab} \left( {t_n - \frac{D_{ab}}{2k^2} }\right)^2 + d_m \right)^\frac{\alpha}{2} dt_n\\ $$ and $$ d_m^{\frac\alpha2}\cdot\left(\...
1
vote
1answer
47 views

Need help with integral involving square root (definite or indefinite version)

I am trying to find an analytical solution for this problem: $ \int_{-a/2}^{a/2}\frac{z}{z^{2}+(x-x')^{2}}\frac{sx'-y}{\sqrt{z^{2}+(x-x')^{2}+\left(sx'-y\right)^{2}}}dx' $ where $x$, $y$, $z$, $a$ ...
2
votes
2answers
81 views

Integration of Associated Legendre Polynomial

I am interested in the following integral $$I=\int_{-1}^1P_\ell^2(x)P_n(x)\mathrm{d}x,$$ where $P_n(x)$ is Legendre Polynomial of $n$th order, and $P_\ell^2$ is Associated Legendre Polynomial. Any one ...
0
votes
4answers
78 views

How to solve $\int (2x^2 + 1) e^{x^2} \, dx$ using elementary integration and integration by parts?

$$\int (2x^2+1)e^{x^2} \, dx$$ It's part of my homework, and I have tried a few things but it seems to lead to more difficult integrals. I'd appreciate a hint more than an answer but all help is ...
4
votes
0answers
141 views

Simplify $\int_0^\infty \frac{\text{d}{x}}{e^x+x^n}$

I seem to have seen quite a lot of integrals in the form: $$\int_0^\infty \frac{\text{d}x}{e^x+(1+x^n)}$$ But none of those hold a closed forms (at least to my knowledge) $$\Large\color\red{\int_0^\...