Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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Compute $F(x)$ and $\displaystyle{\lim_{x \to \frac{\pi}{2}}} F(x)$

Consider the function F defined on $]-\frac{\pi}{2}; \frac{\pi}{2}[$ by $$F(x) = \int_0^x t \tan^2{t} dt$$ Compute $F(x)$ and $\displaystyle{\lim_{x \to \frac{\pi}{2}}} F(x)$ Determine the sign of $F(...
vitalmath's user avatar
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0 answers
26 views

Evaluate the integral $\int_{\gamma} \frac{z^2+1}{(z+1)(z+4)}dz$

Evaluate the integral $\int_{\gamma} \frac{z^2+1}{(z+1)(z+4)}dz$ if $\gamma =\beta +[4\pi ,0]$ where $\beta (t)=te^{it} $ for $ 0<=t<=4\pi $ my attempt: $\int_{\gamma} \frac{z^2+1}{(z+1)(z+4)}...
Confused's user avatar
0 votes
0 answers
28 views

Closed form of ${_2}{F}_1\left(\frac{1}{2},s,\frac{3}{2};-\frac{1}{a^2}\right)$

While solving an integral, I came acorss the term $$ \tilde{I}(a,s)= {_2}{F}_1\left(\frac{1}{2},s,\frac{3}{2};-\frac{1}{a^2}\right).$$ To be precise, it came in the following calculations \begin{align*...
Sam's user avatar
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16 views

What is the minimum about the function $u$ we must have to use Green's formula?

In Evans' book Partial Differential Equations, appendix C.2 Theorem 3, Green's formulas are established: Let $u, v \in C^{2}( \overline{U})$. Then $$ \int_U Du \cdot Dv dx = - \int_U u \Delta v dx + \...
Ilovemath's user avatar
  • 2,901
-1 votes
0 answers
31 views

Integration of time dependent function

Assume that there is a time-dependent function, $\beta(t)$, is it safe to write: $$ \int_0^t{\beta(s)}ds = \beta(t) -\beta(0) $$ Also, if $\beta$ follows a stochastic path satisfying: $$ d\beta(t) = \...
coffee-raid's user avatar
0 votes
4 answers
33 views

Clarification on Simplification of a radical

I recently solved the following integral: $$ \int _1^{\sqrt{3}}\frac{\sqrt{1+x^2}}{x^2}dx $$ After integrating, I obtained the result: $$ \frac{1}{2}\ln\frac{2-\sqrt2}{2-\sqrt3}+\frac{1}{2}\ln\frac{2+\...
SAQ's user avatar
  • 359
2 votes
0 answers
22 views

How to use stationary phase method when the zero point is at infinity?

Thank you for reading my questions. It is known that the stationary phase method has this form: $$ \int_a^bg(t)e^{jf(t)dt}\approx\sum\limits_{t_0\in\Sigma}g(t_0)e^{jf(t_0)+j*\text{sign}(f''(t_0))\frac{...
Xiangyu Cui's user avatar
2 votes
1 answer
81 views

How to integrate with respect to $\overline{z}$

Suppose we want to evaluate $$\int_C f(z) \, d \overline{z}$$ (I'm intentionally being vague with how $f$ and $C$ are defined because my question doesn't focus on these details - see example below for ...
Grigor Hakobyan's user avatar
3 votes
1 answer
62 views

Solve trig integral $\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx $ [duplicate]

I've stumbled across this integral: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx $$ I was on a time limit and my intuition told me: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)...
Avgustine's user avatar
  • 133
3 votes
1 answer
53 views

How to compute $\int\frac1x\Bigl(\sum\limits_{n\ge0}\frac{x^{2n}}{2^{2n}(n!)^2}\Bigr)^{-2}\mathrm dx$?

The background of $$\int\frac{1}{x}\left(\displaystyle\sum_{n\geq 0}\frac{x^{2n}}{2^{2n}(n!)^2}\right)^{-2}\text dx$$ is that it yields a second solution to $xy''+y-xy=0$ if you multiply it by the ...
Joan S. Guillamet F.'s user avatar
3 votes
0 answers
49 views

Two formulations of Riesz–Markov–Kakutani representation theorem

Let $ X $ be a locally compact Hausdorff space, and $ C_c(X) $ be the space of all complex-valued continuous functions with compact support on $ X $. As far as I know, there are two formulations of ...
o-ccah's user avatar
  • 874
-3 votes
0 answers
20 views

\int\int _R(x^2+y^2)^(-2)dA; where R is the region inside the circle x=<1 and x^2+y^2=2. [closed]

\int\int _R(x^2+y^2)^(-2)dA; where R is the region inside the circle x=<1 and x^2+y^2=2.
Ujin Kim's user avatar
1 vote
2 answers
105 views

How to solve $\int_0 ^1 \left( \frac{x^2}{1+x^2} \right)\frac{1-x\tan(x)+ \tan(x)-x}{1 -x\tan(x)-\tan(x)-x}dx$?

I saw this interesting problem: $$\int_0 ^1 \left( \frac{x^2}{1+x^2} \right)\frac{1-x\tan(x)+ \tan(x)-x}{1 -x\tan(x)-\tan(x)-x}dx$$ I tied all the tricks that I know and non of them were useful at ...
pie's user avatar
  • 3,457
0 votes
1 answer
30 views

Measure Theory: Proof regarding a measurability of a function [closed]

Is my proof for the following problem correct? Let $f:X\rightarrow[0,\infty)$ and $X=\bigcup_{i\in\mathbb{N}}A_i$ be measurable. Prove: $f$ is measurable $\iff$ $f\cdot\chi_{A_i}$ is measurable. Proof:...
LGHR's user avatar
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6 votes
0 answers
107 views

How would one integrate over $SO(n)$?

Suppose you want to find the average of an $n\times n$ diagonal matrix $A$ over all possible rotations, $$ \langle A\rangle = \int\limits_\text{SO($n$)} Q^T A Q \; dQ. $$ It's easy enough to do this ...
Xander's user avatar
  • 61
0 votes
0 answers
30 views

Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$

I want to prove the inequality with analysis methods instead of method of PDE Actually we can consider the following PDE: $$\left\{\begin{array}{l} u''(x)=u''(x)\\ u(0)=u(1)=0 \end{array}\right.$$ ...
Gang men's user avatar
  • 425
-4 votes
0 answers
47 views

Does the integral of a quotient equal the quotient of integrals? [closed]

Are the following equivalent? $$\int_{a}^{b}\frac{f(x)+g(x)}{h(x)+j(x)}dx=\frac{\int_{a}^{b}(f(x)+g(x))dx}{\int_{a}^{b}(h(x)+j(x))dx}$$.
Bean6618's user avatar
-2 votes
0 answers
24 views

Is the following statement about integration of a distribution function true? [closed]

Assume $a,b\in\mathbb{R}$ with $a<b$. Assume $F:\mathbb{R}\to[0,1]$ to be a cumulative distribution function. Is the following statement true? $$b-a=\int_a^b dy\geq \int_a^b F(y)dy$$ My ...
mathCurious's user avatar
0 votes
1 answer
27 views

How do I combine a PDF with another continuous function and then do a sum product?

I haven't done math in awhile so am a bit fuzzy on how to set up the problem. Let's say there's a test that students take and they can score anywhere between 0 and 10000. Let's say the distribution of ...
we_are_all_in_this_together's user avatar
0 votes
0 answers
21 views

Theorem 7.20 in Apostol's MATHEMATICAL ANALYSIS, 2nd edition: The Comparison Theorem for Riemann-Stieltjes Integrals

Here is Theorem 7.20, in Chap. 7, in the book Mathematical Analysis - A Modern Approach To Advanced Calculus by Tom M. Apostol, 2nd edition: Assume that $\alpha \nearrow$ on $[a, b]$. If $f \in R(\...
Saaqib Mahmood's user avatar
1 vote
2 answers
138 views

Jeffrey's integrals

I was reading G. B Jeffrey's article with The Royal Society as publisher, and I came across two integrals, for which there were solutions but no procedure how they were obtained. I was trying to do it ...
Edward Henry Brenner's user avatar
0 votes
0 answers
44 views

Deriving the CDF and PDF of the log normal distribution

We have that Y = $e^x$ and that X is a random variable with $X∼N(μ,σ^2)$. I want to obtain the density function $f_Y(y)$ of $Y$, by finding the cumulative distribution function $F_Y(y)$, and then ...
Markus J. Blicher's user avatar
4 votes
1 answer
105 views

Integral $\int_0^{1/\sqrt{2}}\frac{K\left[\sqrt{1-k^2}\right]}{\sqrt{1-2k^2}\sqrt{1-k^2}}dk$

$$\int_0^{1/\sqrt{2}}\frac{K\left[\sqrt{1-k^2}\right]}{\sqrt{1-2k^2}\sqrt{1-k^2}}dk=\frac{\Gamma^2(1/8)\Gamma^2(3/8)}{32\pi}$$ With $K$ as the Complete Elliptic Integral of the First Kind, $$K:=K(k)=\...
Miracle Invoker's user avatar
1 vote
0 answers
66 views

Prove that $\det($Hilbert Matrix minus $\frac{1}{(n+1)^2}$ in the top left$)=0$

If $f$ is a polynomial of degree $n$ such that $\int_0^1 f(x)x^kdx=0$ for all $k=1,2,\dots,n$, then $\int_0^1 f^2(x)dx=(n+1)^2\left(\int_0^1 f(x)dx\right)^2.$ This implies that $\begin{vmatrix} 1-\...
Andyqian7's user avatar
-1 votes
0 answers
18 views

Can you use trig substitution to integrate a function when the sides of the triangle are defined with trig expressions? [closed]

I know the integral of 1/(1-sinx) can be solved by mutiplying by 1 in the form of (1+sinx)/(1+sinx) but what if you made a right triangle and defined the legs as rad(1-sinx) and rad(sinx) with ...
Gallia Vickery's user avatar
2 votes
1 answer
42 views

Integral of the fundamental solution of the laplace equation over a circle

I asked a similar but different question earlier here (this is not a duplicate) I'm interested in solving this problem in closed form, if such a solution exists, in $2d$. $$\int_{\Gamma_R} G\; d\...
Cedric Martens's user avatar
2 votes
0 answers
31 views

Integral of the normal derivative of the fundamental solution of the laplace equation over a circle

I'm interested in solving this problem in closed form, if such a solution exists, in $2d$. $$\int_{\Gamma_R} \frac{\partial G}{\partial \vec{n}} d\Gamma_R$$ Where $\Gamma_R$ is a circle of radius $R$ ...
Cedric Martens's user avatar
1 vote
1 answer
38 views

Fourier transform of $\exp(-\cosh(x))$

I am trying to compute the Fourier transform $$ \hat{g}(\xi) = \int_{\mathbb{R}} e^{-\cosh(x)}e^{-2i\pi \xi x} dx $$ of the function $g(x)=\exp(-\cosh(x))$. Are there any standard substitution tricks ...
mrry0's user avatar
  • 41
-2 votes
0 answers
50 views

A difficult integrate question for me(help! [closed]

enter image description here $$\int\frac{\sin{x}\left(\sqrt{x+1}-1\right)}{\sqrt{1+x^2}\int_{0}^{x^2}{\tan{t^2}dt}}dx$$
xon's user avatar
  • 1
1 vote
0 answers
13 views

Reverse inequality for Riemann-Liouville fractional integral

Definition 2.3. The Riemann-Liouville fractional integral operator of order $\alpha \geq 0$, for a function $f \in C_\mu,(\mu \geq-1)$ is defined as $$ \begin{aligned} J^\alpha f(t) & =\frac{1}{\...
Grandes Jorasses's user avatar
2 votes
0 answers
107 views

Find $f'(1)$ given $\int_0^1f(x)(x-f(x))\,dx=\frac1{12}$ [duplicate]

Consider a function $f:[0,1]\to\mathbb{R}$ satisfying $$\int_{\ 0}^1 f(x)(x-f(x))\,dx=\frac{1}{12}.$$ Then find the value of $f'(1)$ After banging my head on the table for half an hour, I assumed the ...
MathStackexchangeIsNotSoBad's user avatar
0 votes
1 answer
38 views

Integrate $e^{2z}$ along the path $\cos(t)+it$ for $t∈[0,\pi/2]$ [closed]

I need to calculate the integral $$\int_{\gamma_1} e^{2z} \, \mathrm dz \quad \text{for} \quad \gamma_1(t) = \cos t + it,\; t \in [0,\pi/2],$$ but I don't think I can use Cauchy's Theorem or the ...
a Ludan's user avatar
2 votes
0 answers
34 views

Riemann zeta function and Bernoulli numbers

According to Wikipedia (https://en.wikipedia.org/wiki/Bernoulli_number) we have: $$1^m + 2^m + ... n^m = S_m(n) = \frac{1}{m+1} \sum_{k=0}^{m} \binom{m+1}{k} B_k^{+} n^{m+1-k}$$ where $B_k^{+}$ are ...
Nikolai riber skånstrøm's user avatar
3 votes
2 answers
88 views

How to calculate: $ \int_{-7 \pi}^{2 \pi} \frac{1}{2 \sin(x) - \cos(x) + 5} dx$

How to calculate the value of this Riemann integral? $$ \int_{-7 \pi}^{2 \pi} \frac{1}{2 \sin(x) - \cos(x) + 5} dx $$ I used the universal trigonometric substitution and found the following ...
Jacobs Monarch's user avatar
0 votes
0 answers
19 views

Integral of $\cos(\sum_i a_i cos(b_ix))$ as a sum of Bessel J functions

Using Jacobi-Anger expansion, we can represent the below integral in the l.h.s as an infinite sum of Bessel J functions in the r.h.s: \begin{equation} \int_0^L \cos(a\cos(bx)) dx = LJ_0(a) + 2\sum_{n=...
user185671631's user avatar
-5 votes
0 answers
31 views

Solve this exercise plz [closed]

enter image description here Probability
Aladin Hd's user avatar
-1 votes
0 answers
45 views

solve $\frac{d^2x}{dz^2} = -2e^x$ [closed]

Help me to solve $\frac{d^2x}{dz^2} = -2e^x$ I request for help. Thanks
Mon's user avatar
  • 31
3 votes
1 answer
158 views

Remarkable logarithmic integral $\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} dx$

Question: how to evaluate this logarithm integral? $$ I=\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x $$ My attempt: $$ \begin{aligned} I=&\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x\\ ...
ARYAN1201's user avatar
  • 117
0 votes
0 answers
44 views

How to simplify this integral

$$\int_{0}^{\infty} e^{i 2m x - i\Omega \sin^{-1}(\tanh(x))} \,dx$$, whose solution is given to be $$4i\pi m e^{-i\pi(\frac{2m + i\omega}{2})}cosech(\pi\omega) {}_2F_1(1-2m, 1-i\omega, 2; 2) $$, can i ...
Waheed Dar's user avatar
0 votes
1 answer
45 views

Confusion over why Green's Theorem is usable in one situation with closed circle and not the other.

As I understand it, if a region contains the origin, Green's Theorem cannot be applied. However, one question in my issued lecture notes appears to contradict this and another one appears to follow ...
Lim Min Kang's user avatar
0 votes
0 answers
36 views

How to prove this difficult double integral identity?

In this MathOverflow link -originally posted here in MSE- I have placed 4 fast hypergeometric-type series for logarithm constants. I could prove 3 of them by using a Beta function approach ...
Jorge Zuniga's user avatar
0 votes
0 answers
34 views

Computation of integrals with Airy function

Let $\mathrm{Ai}$ denote the Airy function. Is it possible to compute explicitly (or bound) $$ \int_0^{K} \frac{\mathrm{Ai}(z)}{\mathrm{Ai}(-z)} dz $$ for $K >0$ (or also for $K = \infty$)?
zelda's user avatar
  • 1
5 votes
2 answers
100 views

how to evaluate $ \int_0^1 \frac{\arctan \left(x \pm \sqrt{x^2+1}\right)}{x+1} d x $

how to evaluate $$ \int_0^1 \frac{\arctan \left(x \pm \sqrt{x^2+1}\right)}{x+1} d x $$ Denote: \begin{align*} I &= \int_0^1 \frac{\arctan \left(x+\sqrt{x^2+1}\right)}{x+1} \,dx \\ J &= \int_0^...
ARYAN1201's user avatar
  • 117
0 votes
0 answers
24 views

Parameter Dependence Integrals

Let $(X,\mathcal{X},\lambda)$ be a measure space and $(\Theta,d)$ a metric space. Assume a function $f:X\times\Theta\to \mathbb{R}$ be uniformly bounded, meaning $\sup_{x\in X,\theta\in\Theta}|f(x,\...
Mathmaxis's user avatar
0 votes
1 answer
33 views

Why is $2\sin\theta$ the upper limit for this inner integral, and why is $0$ the lower limit?

I am having difficulty understanding how this practice question is supposed to be solved. Here it is: Use polar coordinates to find the exact value of the double integral, $\iint_D x$ $dA$ where $D$ ...
Lim Min Kang's user avatar
7 votes
1 answer
253 views

Evaluating $\int_0^{\pi } \frac{\sin (n \sigma )}{(a-\cos \sigma )^2} \, d\sigma$

What is the formula for $$\int_0^{\pi } \frac{\sin (n \sigma )}{(a-\cos \sigma )^2} \, d\sigma$$ where $ a>1 $ and $n$ is a positive integer? To evaluate, I tried to replace $a$ with $\cosh\xi $ in ...
Rajai's user avatar
  • 73
0 votes
1 answer
43 views

Nice application of dominated convergence theorem

Let $\delta \in \mathbb{R}$, $$f(x)=\frac{sin(x^2)}{x}+\frac{\delta x}{1+x}.$$ Show that $$\operatorname{lim_{n\to \infty}} \int_{0}^{a}f(nx)=a\delta$$ for each $a>0.$ I am unable to find ...
Infinity's user avatar
  • 626
0 votes
0 answers
67 views

Integration by parts does not work for this complex integral. Why?

There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted: $$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\...
Тyma Gaidash's user avatar
1 vote
1 answer
48 views

What is the average of $n(100-n)$ for $n \in {1,2,...,100}$

What is the average of $n(100-n)$ for $n \in {1,2,...,100}$ I saw this problem on the internet without a method of solving it without brute force or a computer. The answer is 1666.5. If you were to ...
Xerium's user avatar
  • 23
2 votes
2 answers
70 views

How to calculate the derivative of a Newtonian potential inside a box with uniform source distribution?

I'm working on a potential flow problem. I have a box, centered at the origin (i.e. $V=[-x_b,x_b]\times[-y_b,y_b]\times[-z_b,z_b]$) that has inside of it a uniform distribution of source strength. We ...
byl's user avatar
  • 55

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