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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2
votes
1answer
46 views

Integrate $\sin^n{x}$

How do you integrate: $\int(\sin^n{x}) dx$ The link to WolframAlpha : (Integration Answer) No definite limits... What is that hypergeometric function in that answer. Please help! Thanks
2
votes
2answers
39 views

trouble solving the integral of $\cos(x^2)$

No, I really mean the integral of $\cos(x^2)$, not $[\cos(x)]^2$. Can the chain rule be applied here?
2
votes
1answer
28 views

Half-Fourier transform, relation to Delta function

so the Fourier transform of the Kronecker Delta function is (up to sign conventions / normalisation) $$\int_{-\infty}^\infty dt\; e^{i t \omega} = \delta(\omega).$$ Can one say anything about the ...
2
votes
2answers
58 views

Integrate $\csc(x)^3dx $ [duplicate]

I'm not sure trig identity to use. Would we use $1+\cot^2(x) = \csc^2(x)$? I know that we break down into $\csc^2(x)$ and $\csc(x)$. Could I get hints?
1
vote
0answers
8 views

Mathematical literature for Dirichlet-multinomial integration

I am playing with topic models (Latent Dirichlet Alocation + modifications) and would be grateful for pointers to mathematical analysis literature that covers details of the derivations. For ...
2
votes
0answers
13 views

Relation between upper sum & Riemann Integration with inequality

Let, $f$ is differentiable function on $[a,b]$ and $f'$ is bounded on $[a,b]$.Then prove that, $|U(P,f)-\int_a^b f(x) dx|<=(b-a)\sup\{|f(x)|:x \in [a,b]\}$, where U(P,f) denotes the upper Riemann ...
3
votes
1answer
45 views

Why do we care about the 'rapidness' for convergence?

It is those puzzeling improper integrals that I can't get my head around.... Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has ...
13
votes
3answers
249 views

Prove $\int_{0}^{\pi/2} x\csc^2(x)\arctan \left(\alpha \tan x\right)\, dx = \frac{\pi}{2}\left[\ln\frac{(1+\alpha)^{1+\alpha}}{\alpha^\alpha}\right]$

When I showed to my brother how I proved \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2 \end{equation} using the following theorem by ...
1
vote
0answers
73 views

Is there a simple proof that $\int_0^1 \lceil f(x) \rceil \mathrm{d}x \geq \int_0^1 f(x) \mathrm{d}x$? [closed]

I want to prove that: $$\int_0^1 \lceil f(x) \rceil \mathrm{d}x \geq \int_0^1 f(x) \mathrm{d}x$$ Is there a simple proof that this is true? The purpose of this proof is for applying formal methods ...
0
votes
0answers
16 views

asymptotic esimation of a complex integral

I am searching for a general method to evaluate asymptotically this kind of integral $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(q,\omega)\exp[-\mathrm{i}kr]\exp[-\mathrm{i}\omega ...
-1
votes
2answers
61 views

Integrate by parts: $ \int x\,\tan^2(2x) dx $

How can I integrate this by parts? $$ \int x\,\tan^2(2x) dx $$ PS: Please do not include any integration with the method of changing variable. Thank you.
2
votes
1answer
44 views

Line integral along Circle

Can you please give me a hint so that I can get started with the following integral? $$\oint_{C(R)} \frac{\sin(\pi/z)}{(1+z)^2} dz$$ where $C(R)$ is a circle with radius $R$ and origin $0$. ...
0
votes
0answers
32 views

How to prove that the integration contour of one integral is the subset of the other integral? [closed]

I have the two integrals which have the same non negative integrand. For example the following two integrals, $\int_{-l}^{l}f(x)dx$ ....... (1) and $\int_{a}^{b}f(x)dx$ ....... (2) What we ...
0
votes
2answers
38 views

How to find $\left\lfloor\sum_{r=1}^{80}\int_0^1x^{\sqrt r-1}dx\right\rfloor$

$$\left\lfloor\sum_{r=1}^{80}\int_0^1x^{\sqrt r-1}dx\right\rfloor$$ My try: $$K=\left\lfloor\sum_{r=1}^{80}\int_0^1x^{\sqrt r-1}dx\right\rfloor=\left\lfloor\sum_{r=1}^{80}\frac1{\sqrt ...
2
votes
1answer
45 views

Having trouble with simple integral

sorry I'm having some trouble evaluating this integral $\frac{dv}{dt} = -k(v-gt)^2-g$ where g and k are constants I'm assuming you just separate and integrate but I cannot seem to get it to work ...
0
votes
0answers
30 views

How do I solve this calculs problem [closed]

a) Find the general solution of $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 0.$$ b) Solve $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 8\cos 2t + 6\sin 2t.$$ with $y(0) = 4$, $y'(0) = 0 $ How ...
1
vote
2answers
42 views

Integrals: Partial Fractions

$$ \int \frac{x^2-x+12}{x^3+3x} $$ I factored the denominator to get $ x(x^2+3) $. I then seperated the x and the $x^2+3$ into the partials $\frac{A}{x}$ and $\frac{Bx+C}{x^2+3}$. After combining ...
1
vote
0answers
27 views

How to parametrize the volume of the intersection of cube and a right tetrahedron?

This is an extension of my previous question. I am trying to find the volume of the region which is the intersection of a cube given by $\vec r_1 = (x,y,z)$, where $$\begin{cases}0 \le x \le 1 \\ 0 ...
1
vote
0answers
11 views

Create equal areas with the intersection of a function with the first and fourth quadrants.

Here's a question I wasn't able to figure out in class. The first part was easy enough, but the second part stumped even my teacher as well. "Consider the first- and fourth quadrant regions between ...
2
votes
1answer
26 views

Integration by parts with c value?

Ive asked this question a few times but still don't understand how to go about and there have been a few answers that a different so can anyone clarify what the correct answer for this integral would ...
0
votes
0answers
14 views

Integration by parts with Legendre Functions

I need help deriving $\int_{-l}^l [P_l^m(x)]^2 = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$ for the associated Legendre functions I am supposed to use $P_l^m(x) = (-1)^{-m}\int_{-l}^l ...
1
vote
1answer
11 views

Bound of an integral of a bounded function

If a function $f$ is bounded such that $|f|<M$, then can we say: $$ \left|\int_a^ \infty f(x)\, \mathrm{d}x \right| \le \int_a^ \infty |f(x)| \,\mathrm{d}x \le \int_a^ \infty M\, \mathrm{d}x$$ ...
0
votes
1answer
27 views

Quadratic equation form

I have the relation $u=\sqrt{(a_1+b_1t)^2+(a_2+b_2t)^2+(a_3+b_3t)^2} \tag 1$ I need to write $t$ as a function of $u$ ($t=f(u)$). How will I get that ? NB: $a_1,a_2,a_3,b_1,b_2,b_3$ are ...
4
votes
1answer
39 views

Why can I not combine integrals this way?

Evaluating the triple integral $\int^1_0 \int^{1-z}_0 \int^{1-y-z}_0 \text{dxdydz}$, I get $\frac 16$. Evaluating the triple integral $\int^1_0 \int^1_0 \int^1_0 \text{dxdydz}$, I get $1$. So I ...
1
vote
0answers
20 views

Determine whether $\lim_{R\to\infty}\int_0^R\frac{|\sin x|}{x}dx-\frac{2}{\pi}\ln R$ exists

Let $$J(R):=\int_0^R\frac{|\sin x|}{x}dx.$$ (i) Show that $$\lim_{R\to\infty}\frac{J(R)}{\ln R}$$ exists and determine its value (ii)Does $$\lim_{R\to\infty}J(R)-\frac{2}{\pi}\ln R$$ exist? If ...
0
votes
2answers
24 views

Calculus find limit with dominance of one function over another

so I have this math problem, that I can't seem to wrap my head around, I have to find the integral: $$\int_1^{\infty}{\frac{d}{dx}}\left(\frac{3\ln(x)}{x}\right)dx$$ I have no idea where to start... ...
0
votes
0answers
21 views

Differential equation Worded Problem [duplicate]

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
2
votes
2answers
27 views

Find $\int (x^3+1)(\cos(2x)) dx$

What I've tried for this problem is expanding it to $x^3\cos(2x) + \cos(2x)$ and then evaluating the respective functions as separate integrals. The first one uses tabular and the second one is simple ...
0
votes
1answer
18 views

Differential equation with modulus

I have a problem with $$y-xy'=(ln|x|+1)y^2 $$ because I do not know how to deal with the absolute value. I divide $\frac{y - xy'}{y^2}=ln|x| +1$, then substitute $t' = (\frac{x}{y})'$ get ...
5
votes
3answers
134 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
1
vote
3answers
30 views

Calculus find out if integral converges or diverges

So I have this math problem, where I am supposed to find out whether or not the integral converges or diverges and solve. $$\int_0^1 \frac{3\,dx}{\sqrt{x}(x+1)}$$ I'm not 100% sure as to figure out ...
2
votes
2answers
33 views

Integration by parts polynomial times logarithim

$$\int \sqrt{x}\log_2(x) \, dx$$ Integral of Square root of x times log base 2 of x dx.
8
votes
4answers
132 views

Evaluate $\int(x^{91}+x^{327})\cos(x)\mathrm{d}x \quad .$

Evaluate $$\int\left(x^{91}+x^{327}\right)\cos(x)\mathrm{d}x \quad .$$ It's my first time to face integration like that. I just need a clue to start because I tried, but it's not working Thanks in ...
0
votes
1answer
21 views

Calculating the limit of the “$\dfrac{volume}{area}$” ratio for a 2D function

Let's assume that we have a well behaving, continuous function $f(x,y)$ defined on $\mathbb{R^2}$. The double integral $\int_{x_0}^{x_1}\int_{y_0}^{y_1}f(x,y)dxdy$ gives the volume of the space ...
1
vote
2answers
25 views

How to integrate $(x-1)^4/(x^2 )$?

How to integrate $\frac {(x-1)^4} {x^2 }$ ? I really tried hard but don't know how to start please guide me to just start thanks in advance
0
votes
2answers
34 views

Spectral Measures: Support vs. Norm

Given a complex Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: $$T:=\int_\mathbb{C}zdE(z)$$ ...
0
votes
3answers
39 views

Spectral Measures: Support vs. Spectrum

Given a complex Hilbert space $\mathcal{H}$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: ...
0
votes
1answer
12 views

Does a constant of integration changes the shape of a distribution?

Let $f(x)$ be the frequency distribution of the variable $x$. Let assume that $\int^{\infty}_{-\infty} f(x) ≠ 1$. Let $g(x) = C f(x)$ such as $C$ is the constant of integration so that ...
1
vote
2answers
47 views

How to find $\int_0^{\pi/4}\sec(x)^3dx$

How do I find $$\int_0^{\pi/4}\sec(x)^3dx$$ I arrive at $\sec x\tan x + 1/3 \ln| (\cos^3(x))| dx$ where $u = \sec x$ and $v' = \sec^2(x)$ What's my error? Could I get a step by step solution?
2
votes
0answers
57 views

Integral with cosines and power or upper bound

I need to solve this integral or find its upper bound $$\int_M^\infty \frac{2}{(t^2 \pi^2 + \epsilon^2)^\beta}\sin(\pi x t)\sin(\pi z t)\mathrm{d}t$$ I got to simplify upstairs as $$\int_M^\infty ...
1
vote
2answers
71 views

Linear functionals and dual bases

How do I tackle this question? I am a little hazy on linear functionals and integral signs.
0
votes
2answers
34 views

Green's theorem exercise

I am trying to solve the following problem: Show functions $P,Q:\mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$ of class $C^1$ that verify $P_y=Q_x$ but $$\int_\gamma P(x,y)dy+Q(x,y)dy \neq 0$$ where ...
0
votes
1answer
34 views

Find the resolvent kernel of particular form of volterra integral equation of second kind

Find the resolvent kernel for the integral equation $$u(x)= 30 +6x+∫_0^x(5t-6t^2 )u(t)dt$$ I try to solve but its lengthy $$k(x,t)=(5t-6t^2)$$ $$k_2(x,t)=∫_ξ^x (5t-6t^2)(5ξ-6ξ^2)dt$$ its very ...
1
vote
0answers
26 views

Integration by substitution of reciprocal of polynomial times logarithim

$$\int \frac 1 {\log_4^2 (x)} dx $$ I used $u = \log_4(x)$ and arrived at the solution $-\ln(4)/(\log _4 (x)) $, but I think this is wrong. How should I do it correctly?
1
vote
2answers
149 views

How to integrate by parts, without changing variable

How can I solve: $$\int \frac{x^2}{\sqrt{1-x^2}}\;dx$$ without changing variables, by parts?
1
vote
1answer
31 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
0
votes
0answers
11 views

how to Double integrate Using dblquad command [closed]

./2))/(pi*(y_bar./(2*(1-x_bar.^2).^0.5))./2)).^0.5)(1./cos(pi(y_bar./(2*(1-x_bar.^2).^0.5))./2))((0.752)+(2.02(y_bar./(2*(1-x_bar.^2).^0.5)))+((0.37)(1-sin(pi(y_bar./(2*(1-x_bar.^2).^0.5))./2)).^3))); ...
15
votes
2answers
188 views

Limit of $\int_0^1\frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
1
vote
1answer
27 views

How to calculate the Gaussian Integral in specific region?

Firstly, I know that the Gaussian Integral formula, e.g., $\int^{+\infty}_{-\infty}e^{-ax^2}dx=\sqrt{\frac{\pi}{a}}$. But, I am now being encountered a problem when the integral region is not ...
4
votes
4answers
100 views

Proof of formula for the antiderivative of $\sec x$ [duplicate]

How do I prove that indefinite integral of $\sec x$ is equal to $\ln(\sec x + \tan x) + C$? I tried to substitute $t = \cos x$ but that didn't help. I have no idea how to integrate it any other way, ...