All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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0
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2answers
36 views

Resolution of limits like this: $\lim_{n\to+\infty}\int_{0}^{\frac{\pi}{2}} e^{-n \sin(x)} \,dx $

Good morning. Can you give an help to solve these limits? I have thought of using the uniform convergence $$a)\lim_{n\to+\infty}\int_{0}^{\frac{\pi}{2}} e^{-n \sin(x)} \,dx \\ ...
-1
votes
1answer
31 views

Evaluate the limit integral

I have tried to use the Lebesgue Dominated Convergence Theorem to evaluate: $$\lim_{n\rightarrow \infty} \int_{(0,1]} f_n \;d\mu $$ with $f_n(x)=\dfrac{n\sqrt{x}}{1+n^2x^2}$ and ...
5
votes
3answers
72 views

How to integrate $f(x) = \frac{1}{a + b \cos x + c \sin x }$ over $x \in (0,\pi/2)$

Conjecture 1 $$ \begin{align*} I_{T}=\int_0^{2\pi} \frac{\mathrm{d}x}{a + b \cos x + c \sin x} & = \frac{2\pi}{\rho} \tag{1} \\ I_{T/4} = \int_0^{\pi/2} \frac{\mathrm{d}x}{a + b \cos x + ...
0
votes
2answers
35 views

Inequality in Integral

Show that $\dfrac{28}{81}<\int_0^\frac{1}{3}e^{x^2}dx<\dfrac{3}{8}$. It would be great if a solution based on the Mean Value Theorem for Integrals is posted.
0
votes
1answer
25 views

For what functions $f$ does the following integral equation hold?

\begin{equation}\int_{-\infty}^{\infty} f \ dx = \frac{1}{2\alpha} \int_{-\infty}^{\infty} \left( \int_{x - \alpha}^{x + \alpha} f \ dx'\right) dx \end{equation} To show that there is at least one ...
0
votes
2answers
34 views

Integration problem

I'm having trouble with this integration for some reason: $$\int\left(2\cos(2x)+2\right)^{3/2}\mathrm dx$$ Anyone know a quick and easy method to solve this?
0
votes
1answer
27 views

Hard integral of root function and hyperbolic function

I need to calculate this integral: $$ \int^A_B{ \frac{\sqrt{x-B} } { \cosh(x)^2 } } $$ Is there any way to do this?
1
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3answers
101 views

Demonstration of $\int_{a}^b f(x) \,dx= 0 \Rightarrow f(x)\equiv0 $ [on hold]

Good morning, Can you give me a help to demonstrate this proposition: $f$ is a continuous and not negative function on the interval $[a,b] \ a,b \in \Re $, Demonstrate: $$\int_{a}^b f(x) \,dx= 0 ...
2
votes
1answer
23 views

Fourier transform of a divergent function

During a calculation of Feynman diagram, I encountered an integral which is diverging: $$\int d^{2}p\frac{p^{2}p_{n}}{\left(p^{2}+\alpha k^{2}\right)^{2}}e^{ip\cdot y}$$ where $p$, $k$ and $y$ are ...
0
votes
1answer
19 views

Evaluating a surface integral using symmetry

Let $S$ denote the surface of the cylinder $x^2+y^2=4,-2\le z\le2$ and consider the surface integral $$\int_{S}(z-x^2-y^2)\,dS$$ How can one use geometry and symmetry to evaluate the integral without ...
1
vote
1answer
64 views

How to find $\max|f(z)|$ in complex analysis?

The $M-L$ estimation lemma inequality states: $$\left |\int_\Gamma f(z) dz\right| < ML(\Gamma)$$ Where $M = \max|f(z)|$ and $L(\Gamma)$ is the arc length of $\Gamma$. Here: Wikipedia: ...
2
votes
1answer
51 views

Replacing $\sin(z)$ with $1 - e^{2iz}$

I have seen many integral evaluations within logs where they change the sine to: $$\sin(z) \rightarrow 1 - e^{2iz}$$ Such as here: Contour integral evaluation. I dont understand how those ...
0
votes
1answer
56 views

Nspire cx CAS - Laplace inverse fails

I'm trying to calculate that easy integral but I get undef. When I replaced $\infty$ with $1000$, I got the right answer. ($e^{-1000}$ is zero roughly). Although this calculator knows that ...
5
votes
1answer
57 views

Evaluate $\int_{-2}^{-1}\frac{\text{d}x}{\sqrt{-x^2-6x}}$.

Problem statement [from Charlie Marshak's Math GRE Prep Problems]: Evaluate $\displaystyle \int\limits_{-2}^{-1}\dfrac{\text{d}x}{\sqrt{-x^2-6x}}$. My work: notice that $$\begin{align} -x^2-6x ...
2
votes
1answer
77 views

Integration by nonobvious substitutions

The standard technique for evaluating the integral $$\int \sec x \,dx$$ is making the nonobvious substitution $$u = \sec x + \tan x, \qquad du = (\sec x \tan x + \sec^2 x) dx,$$ which transforms the ...
16
votes
4answers
329 views

Integrating $\int_0^\pi \frac{x\cos x}{1+\sin^2 x}dx$

I am working on $\displaystyle\int_0^\pi \frac{x\cos x}{1+\sin^2 x}\,dx$ First: I use integrating by part then get $$ x\arctan(\sin x)\Big|_0^\pi-\int_0^\pi \arctan(\sin x)\,dx $$ then I have ...
1
vote
0answers
45 views

Evaluate $\displaystyle\int_{0}^{1} \frac{\log(x)}{\sqrt{1 - x^2}}$ complex integration [duplicate]

Evaluate: $$2\cdot\int_{0}^{1} \frac{\log(x)}{\sqrt{1 - x^2}} dx$$ Using Complex Integration. I want to do something with the unit circle, but I am not quite sure how to work-around with the unit ...
3
votes
2answers
56 views

Integrate $\int_0^\infty \int_0^\infty \frac{\sin \pi x}{(y+e^x|\sin \pi x|)^2}dx \, dy$ using Fubini or Tonelli theorems

I am trying to show that this integral $$\int_0^\infty \int_0^\infty \frac{\sin \pi x}{(y+e^x|\sin \pi x|)^2}dx \, dy $$ exists and is finite and then finding its value. Since $\sin \pi x$ takes ...
2
votes
1answer
29 views

Expanding in a Fourier series $y = |\cos x|$

How to expanding in a Fourier series function $y = |\cos x|$? Especially interested in how to find $$a_n= \frac{2}{\pi}\int\limits_{0}^{\pi}|\cos x|\cos(nx)dx$$
1
vote
3answers
89 views

Integral of $\log(\sin(x))$ using contour integrals

I know the integral is possible with a simple fourier series expansion of $-\log(\sin(x))$ But I am interested in complex analysis, so I want to try this. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
-7
votes
0answers
52 views

Help me integration [on hold]

Let $B=\{(x,y,z)\mid x^2+y^2+z^2\leq a^2\}$, find $$\iiint_B(x^2+y^2)\,dV.$$
0
votes
1answer
22 views

Lebesgue integral, integer part x

$$ \int_{0}^{\infty} 10^{-2[x]} dx $$ How to solve it? is the Lebesgue integral. I drew a graph, it is piecewise continuous. Sum of this function will converge. But I can not understand how it all ...
2
votes
0answers
69 views

Changing the order of integration in the proof that Laplace maps convolution to multiplication

I was reading the proof that Laplace transform maps the convolution of two functions to the multiplication of their transforms. Or mathematically $$\mathcal{L}[f*g]=\mathcal{L}[f]\,\mathcal{L}[g],$$ ...
1
vote
0answers
23 views

Constructing set of functions that give a good basis after a certain integral

I need a set of functions that can be used as a basis after a specific integration. In other words: I integrate a set of functions enumerated by a parameter $k$ where the integral depends on another ...
10
votes
3answers
213 views

What are other methods to evaluate $\displaystyle\int_0^1 \sqrt{-\ln x} \ \mathrm dx$

$$\int_0^1 \sqrt{-\ln x} dx$$ I'm looking for alternative methods to what I already know (method I have used below) to evaluate this Integral. $$y=-\ln x$$ $$\bbox[8pt, border:1pt solid ...
2
votes
0answers
621 views

The partial expectation 𝔼(X|X>K) for an alpha-stable distributed random variable

The partial expectation $\mathbb{E}(X|_{X>K})$ for an alpha-stable distributed random variable: By playing with convolutions of Characteristic Functions of alpha-Stable distributions $S(\alpha, ...
2
votes
1answer
57 views

Complex Contour Integrals from integrals from $0 \to 1$

Evaluate: $$\int_{0}^{1} \frac{dx}{1 + x^3}$$ The bounds are not from 0 to infinity or from -infinity to infinity etc.. How can we use complex contour integration for this? Thanks
4
votes
2answers
198 views

Integral identity?

Is this statement true? If so, I do not know why, can anyone explain please: $$\int_{-\infty}^\infty e^{-(k-x)^2} \,dx=\int_{-k}^k e^{-x^2} \,dx$$ Thank you!
4
votes
1answer
30 views

exists $A \in \mathcal{F}$ such that $\mu(B\triangle A) < \epsilon$

Let $\mu$ be a probability measure on $(S, \mathcal{S})$, where $\mathcal{S} = \sigma(\mathcal{F})$ for a field $\mathcal{F}$. How do I go about showing that for each $B \in \mathcal{S}$ and $\epsilon ...
4
votes
2answers
133 views

Integral along a contour is $0$, how?

I recently had an extremely failed attempt at asking the same question, so I am posting the same question more or less to hope that someone can give me feedback. Consider the integral: ...
2
votes
1answer
70 views

Evaluating $\int_0^{\infty} \frac{\xi x^{\alpha}}{ e^{x}-\xi} \:\mathrm{d}x$

I am supposed to integrate for $\alpha \ge 0$ $$\int_0^{\infty} \frac{x^{\alpha}}{ \xi^{-1}e^x - 1} \:\mathrm{d}x,$$ where $\xi e^{-x} < 1$ which means, I want to express this in terms of simple ...
0
votes
0answers
13 views

application of line integral with repect to x or y.

how line integrals with respect to x or y are used. I'am looking for how they are useful ? what kind of math or real life problems they solve? \begin{align} \int_C f(x,y)\,dx &:= ...
2
votes
0answers
24 views

Interpretation of a line integral with respect to x or y .

i read about some interpretation ideas in Interpreting Line Integrals with Respect to $x$ or $y$ and i was wondering if the interpretation given below is right or not ? lets say we have : ...
5
votes
3answers
166 views

Integrate $I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$

I have a problem with this integral. It seems that solution has to be simple, but I couldn't find out. $$I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$$ I tried using integration by parts and ...
0
votes
0answers
29 views

Showing that $\int_{c} \omega =0$ when $\partial c =0$

Let $\omega$ be a $k$-form on $\mathbb{R}^n$ and suppose that $\omega=d\alpha$ for some $(k-1)$-form $\alpha$. Show that, for any singular $k$-cube $c$ on $\mathbb{R}^n$ with $\partial c=0$, ...
0
votes
0answers
26 views

Using triangle inequality for upper limit of integral.

Prove that $$\left|\int_C(e^z-\bar z)\,dz\right|\le60$$ What I did: $$\left|\int_C(e^z-\bar z)\,dz\right|\le M_CL$$ where $M_C$ is maximum value of $|e^z-\bar z|$ which by triangle inequality ...
1
vote
1answer
27 views

Let $\{f_k\}$ be a sequence of non-decreasing fcns. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu = \int_X \lim_k f_k d\mu$

I need your help to understand and analyse the following problem: Q: Let $\{f_k\}$ be a sequence of non-decreasing measurable function on $(X,\mathcal{A})$ and $\mu$ be a positive measure. If $\int_X ...
1
vote
0answers
13 views

Finding the area under a speed time graph

I recently learned about integration and I wondered how it could be applied to a speed time graph since it does not have a particular equation of a line that one can integrate. Do you split it into ...
3
votes
2answers
50 views

Integral with contour integration

I want to evaluate the integral: $$\int_{-\infty}^{0}\frac{2x^2-1}{x^4+1}\,dx$$ using contour integration. I re-wrote it as: $\displaystyle \int_{0}^{\infty}\frac{2x^2-1}{x^4+1}\,dx$. I am ...
2
votes
0answers
54 views

A Book recommendation for double Integrals?

I have a really hard time learning Double Integrals, which I attempted to understand when I first saw the use of polar co-ordinates for Integrals. So my goal is to learn double Integrals and also ...
1
vote
1answer
53 views

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, ...
0
votes
1answer
54 views

Double Integral Problem - Transforming limits

The double integral $\int_0^2\int_x^{4-x} f(x,y) dy dx $ under the transformation $ u = x+2$, $v = y- 2x$ is transformed into? My attempt: I used the equations $y=x$ and $y=4-x$ to get the relations ...
1
vote
0answers
43 views

Planning to integrate $\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$ using complex analysis [duplicate]

This is just a plan-out. I want to evaluate: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using a keyhole contour a semi-circle, with base at the x-axis. First I must pick a branch. ...
-1
votes
0answers
36 views

Find height h in a cubical box [on hold]

Given a cubical closed box of edge ‘$a$’ cm with liquid filled inside it up to height H. Now some liquid is being removed from the box such that only liquid up to h height is left in the box.Suppose ...
8
votes
3answers
146 views

How to get the solution to these differential equations

I would like to get from $$ \tan(x) = \frac{y''}{y'} + y' $$ The answer is $$ y = \ln(c_1\tanh^{-1}(\tan(\frac{x}{2}))+c_2) $$ The other equation is $$ \sec(x) = \frac{y''}{y'}+y' $$ The answer ...
2
votes
1answer
57 views

Computing the integral $\int\limits_{-\infty}^{\infty}(t^2-1)\delta(t)\:dt$

Have I solved this problem correctly? \begin{align} ...
1
vote
2answers
39 views

Solving system $dx = \frac{dy}{2xz} = -\frac {dz}{2xy}$

How to solve the system $$dx = \frac{dy}{2xz} = -\frac {dz}{2xy}?$$ How in general such systems are solved? Thank you in advance.
5
votes
3answers
126 views

Is the function $f(x) = \begin{cases} 1 & \text{$x\in\Bbb Q$} \\[2ex] 0 & \text{$x\notin\Bbb Q$} \end{cases}$ Riemann integrable?

$f(x) = \begin{cases} 1 & x\in\Bbb Q \\[2ex] 0 & x\notin\Bbb Q \end{cases}$ Is this function Riemann integrable on $[0,1]$? Since rational and irrational numbers are dense on $[0,1]$, no ...
0
votes
0answers
41 views

Contour integration when pole is outside the contour

Here they are using the pole OUTSIDE the contour? I thought this was illegal according to the residue theorem or we are not supposed to do contour integration with poles outside the contour itself.
7
votes
2answers
127 views

Integrate $1/x$ by parts.

$$\int \frac{\mathrm{d}x}{x}$$ If I integrate this by parts ($\displaystyle u=\frac{1}{x}, \mathrm{d}u = -\frac{\mathrm{d}x}{x^2}, \mathrm{d}v= \mathrm{d}x, v = x$), then why does this happen? $$\int ...