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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6
votes
3answers
111 views

How do I compute this integral?

I'm wondering how to compute the integral $$ \int_2^3\int_0^\sqrt{3x-x^2}\frac{1}{(x^2+y^2)^{1/2}}\,\mathrm{d}y\mathrm{d}x. $$ Clearly it is too complicated to do it directly, so I'm guessing you have ...
3
votes
1answer
38 views

Show that $\mu(f)\mu(1/f)\geq\mu(\Omega)^2$

Prove that $\mu(\Omega)^2\leq\int f \,d\mu\int\frac{1}{f}\,d\mu$. I don't know if that what I did is correct or if it will help to solve the problem, but here it is: Using the Hölder inequality ...
4
votes
2answers
60 views

If $\mu(|f_n|^p)$ is bounded and $f_n\to f$ in measure then $f_n\to f$ in $L^1$

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of real measurable functions s.t., (a) The sequence $\displaystyle(\int |f_n|^p\ \mathsf d\mu)_{n\in\Bbb{N}}$ is bounded. (b) The sequence ...
7
votes
1answer
157 views

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\dfrac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$ is a fundamental solution to the biharmonic operator ...
-2
votes
4answers
50 views

How to integrate $\frac{\sec x}{ \tan^2 x}$ [closed]

How to integrate $\frac{\sec x}{ \tan^2 x}$ , already tried dividing it first but am not getting it
0
votes
1answer
67 views

Any suggestions on how could I solve this integral involving a square-root of a polynomial in the denominator?

This is not a homework problem, so there is no guarantee that this integral is solvable analytically. $$ \int_0^\infty \frac{x^2(1-x/2)}{\sqrt{x^2(1-x/2)^2+b}}dx\,. $$ It looks simple enough, but ...
0
votes
0answers
32 views

Fourier series for a Sinusoid in a conventional way?

So my TA in class introduced this amazing way of finding fourier series coefficients for a sin wave, by writing $ sin( \omega t ) = (e^{i\omega t}-e^{-i\omega t}) / 2i $ ----(1) Hence getting the ...
2
votes
2answers
50 views

Find the Integrating Factor

Show that the differential equation $(1-2x^2y^2-4xy^3)dx + (2-2x^3y-4x^2y^2)dy=0$ is not exact, but admits integrating factor $\mu=\mu(xy)$. Find $\mu$ and solve the equation. With the method I ...
0
votes
0answers
21 views

Integral equation solving methods

There is optimization problem, which is about unknown function $\varphi$ under integral sign: $\iint\limits_{[a, b] \times [c, d]}^{} K(x, y) \varphi (x, y) dxdy \to \max$ where is $\varphi (x, y) \in ...
1
vote
3answers
21 views

Transformation to polar coordinates in an integral

Suppose that the domain of integration for a double integral is: $\{(x,y), - \infty < x \le a, -\infty < y \le a \}$. If I want to do a change of variable (to polar coordinates), how do I ...
8
votes
5answers
239 views

Integration of $e^{-x^2}$

I know that this has been asked many times, but I am very interested in the indefinite integral of $e^{-x^2}$. It is stated and proved that there is not elementary way to solve this. So does this ...
3
votes
3answers
142 views

How to integrate $ \int \frac{x^2}{(x \sin(x)+\cos(x))^2} \mathrm{d}x$

Evaluate $$\displaystyle \int \frac{x^2}{(x \sin(x)+\cos(x))^2} \mathrm{d}x$$ Can someone just tell me the necessary manipulations? Hints will be enough. Can it be done by integration by ...
97
votes
19answers
16k views

Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your ...
-1
votes
0answers
22 views

pointwise converges and integrals

Let $(N,P(N),\mu)$ be a measure space such that $\mu(A)=\sum_{n\in A}{1\over n^2} $ a. Let $ f_n = n^2 * 1_{\{n\}} $. Does the sequence converges pointwise? b. Find all functions $ f:N \rightarrow R ...
2
votes
2answers
455 views

Any tips to solve this integral : $I_1 = \int \ln(x^2)e^{\sin(x)}\sin(x^{\cos(x)}) dx$

Background: I was making new expressions to see whether I could efficiently find their derivatives... After having done that, I've started trying to integrate most of them; obviously most of them ...
2
votes
2answers
28 views

Let $f:\Omega\to(0,+\infty)$ and $\ln(x)$ be $\mu$-integrable

Show that $\displaystyle \lim\limits_{p\to 0^+} ||f||_p = \exp(\int\ln(f)\,d\mu)$. In case it comes to be helpful. So far I've shown that $\displaystyle\lim\limits_{p\to ...
2
votes
1answer
63 views

Integral Evaluation.

How can we justify the fact that some integrals can't be evaluated? It's like we can't sum up a function within two bounds or we are unable to find the area under the curve of a function. How's that ...
18
votes
4answers
194 views

How to Prove : $\frac{2}{(n+2)!}\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^{n+2}=\frac{n(3n+1)}{12}$

While I calculate an integral $$ \int\limits_{[0,1]^n}\cdots\int(x_1+\cdots+x_n)^2\mathrm dx_1\cdots\mathrm dx_n $$ I used two different methods and got two answers. I am sure it's equivalent, but ...
-2
votes
1answer
71 views

Integration by parts prove integral of $\cos^n x dx$ [closed]

I'm having a problem with one of my questions. How can I prove that $$\int\cos^n x dx=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx?$$
-2
votes
1answer
54 views

How to find the general solution of the following differential equation [closed]

Could someone please explain to me how to solve the differential equation below: \begin{equation*} 2y\cot x\frac{dy}{dx} = (4+y^2)\cos x? \end{equation*} Thank you very much :)
2
votes
3answers
72 views

how to integrate $\dfrac{\cos x-\cos2x}{1-\cos x}$

I want to find $\displaystyle\int\dfrac{\cos x-\cos2x}{1-\cos x}\ dx,$ I have tried solving the question using substitution. how do I solve it?
0
votes
1answer
66 views

how to integrate (x-1)/(x+1)

I want to calculate the integral $$\int\frac{x-1}{x+1}\,\mathrm{d}x.$$ I have tried solving it by differentiating the denominator and substituting it, but I didn't get it. How else can I solve it?
0
votes
0answers
37 views

How can I find these integral value of $y(t) = 1 - e^{-2t}sin(4t)$?

How can I find... the integral square value the integral absolute value $y(t) = 1 - e^{-2t}sin(4t)$ ? Please help. Thank you
2
votes
3answers
84 views

Finding $\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$

How can I find $$\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$$ ? I suspect this has something simple to do with the basic definite integral properties; ...
1
vote
1answer
41 views

$\int_{0}^{\pi} f(t) \sin (nt)\mathrm dt =0$, for all $n\in\mathbb{N}$ and $f(0)=0$ then show $f \equiv 0$

This question is from a old NBHM Phd scholarship test paper and stuck on it for a long time, here goes the problem: Let $f: [0,\pi] \to \mathbb{R}$ be a continuous function such that $\int_{0}^{\pi} ...
11
votes
7answers
193 views

Where to find interesting integrals for a Calc III student?

I apologize in advance if this is a very soft question. I won't be surprised or offended if I can't get a good answer. One of my favorite things to do in my spare time, when I'm feeling analytical of ...
3
votes
3answers
139 views

Evaluating $\int \frac{\sin\left(x\right)}{1+x^2}dx$

$$\int \frac{\sin\left(x\right)}{1+x^2}dx$$ I have tried to integrate by parts but it doesn't work. How do I evaluate it? Any advice, hint or well-thought solution will be appreciated.
-1
votes
1answer
115 views
0
votes
0answers
26 views

Easy question, hard solution: find the area about a domain in the plane?

We want to find the area of a domain with piecewisely smooth boundary by using the coordinates $(p,\theta)$ of the random line: It has been known that every straight line $\ell$ on $R^2$ can be ...
-6
votes
4answers
73 views

Easy integral with cube root [closed]

Can you help me? evaluate the following integral: $$\int_0^\frac {\pi} {2} \frac {\sqrt[3]{(sin^2x)}} {\sqrt[3]{(sin^2x)}+{\sqrt[3]{(cos^2x)}}}\,\mathrm{d}x$$ Give me show step by step solutions ...
1
vote
0answers
42 views

Finding rate of change with integration (my solution correct?)

The rate of change in a person's body temperature, with respect to the dosage of $x$ milligrams of a drug, is given by $D'(x)=\frac{7}{x+8}$. One milligram raises the temperature 3.7 C. Find the ...
1
vote
1answer
754 views

Change of variables and integral bounds in double integrals

I am preparing for an exam and am having trouble calculating the following integral. $$\iint\limits_B \exp\bigg(\frac{y-x}{y+x}\bigg)\,dx\,dy$$ where $B$ is the the interior of the triangle with ...
0
votes
1answer
62 views

Double integrals for reconstructing probablistic model

I am trying to reconstruct this probabilistic model, \begin{equation} \begin{split} \frac{1}{\mu}\int^{\infty}_{0}P(N \geq n\, |\, L=l, T=t)\,e^{-\frac{l}{\mu}} dl &= ...
0
votes
0answers
56 views

There is some known deficiency through the Lebesgue Integral?

The Integral in the Riemann sense has a lot of deficiencies, and the Lebesgue Integral can solve almost all of them. I know that over limited intervals, Lebesgue Integral is a generalization of the ...
5
votes
1answer
78 views

A difficult double integral

How can the double integral $$ \int_{0}^{\infty}\int_{0}^{\infty}e^ {{-\beta(x^2+y^2+xy)}}\cos(2\pi Mx)\cos(2\pi Ny)\,\text{d}x \,\text{d}y, $$ where $\beta>0$ and $M$, $N\in\mathbb{Z}$, be ...
4
votes
3answers
139 views

Closed-form of $\int_0^1 \operatorname{Li}_3\left(1-x^2\right) dx$

By using dilogarithm functional equations we can show that $$ \int_0^1 \operatorname{Li}_2\left(1-x^2\right)\,dx = \frac{\pi^2}{2}-4, $$ where $\operatorname{Li}_2$ is the dilogarithm function. Could ...
1
vote
0answers
34 views

Does the following integrand have an evaluation?

One of the many terms in an equation I have derived has the following: $\int^{+\infty}_{-\infty} \text{sin}^2\left[a(1+erf(x))\right]H_n(x)\text{exp}(-x^2)dx$ H is the Hermite polynomial and n is an ...
1
vote
2answers
57 views

Show that a function defined by an integral is differentiable

Define $$g(a)=\int_{0}^{\infty}\frac{\sin(ax)}{x}e^{-x}dx,\ \ \ \ \ \ a\in\mathbb{R}$$ a) Show that $g(a)$ is differentiable and compute $g'(a)$. b) Use this to compute $g(a)$. I have tried various ...
0
votes
1answer
36 views

Simpson's rule with precision?

I have an integral: $$\int_0^1sinx^2dx$$ Task is to solve this integral using Simpson's rule with precision $\frac{1}{2}10^{-4}$. I am not sure how should I do that. I have this formula for ...
2
votes
1answer
67 views

Does the integral $\int_0^{\pi} \frac{dx}{sin(2x)+cos(3x)}$ exist?

This link http://www.wolframalpha.com/input/?i=integral%28x%3D0%2Cpi%2C1%2F%28sin2x%2Bcos3x%29%29 shows the visual representation of the integral $$\int_0^{\pi} \frac{dx}{\sin(2x)+\cos(3x)}$$ ...
0
votes
4answers
82 views

Integrating $\frac{1}{(x^4 -1)^2}$ [closed]

How to solve the the following integral? $$\int{\frac{1}{(x^4 -1)^2}}\, dx$$
0
votes
1answer
39 views

Is $lim_{x \to a}\int_{g(x)}^{h(x)}f(t)dt$, where $g(a) = h(a)$, always equal to $0$?

I don't know how to handle such limits. I feel like it should always be equal to $0$, and that's what wolfram alpha says for examples I find, but I'm not sure. I would feel much more secure if someone ...
3
votes
3answers
119 views

Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$

The integral is $$\int \dfrac{(1 + x)\sin x}{(x^2 + 2x)\cos^2 x-(1 + x)\sin2x}dx.$$I've tried the problem by first multiplying both the numerator and denominator by $\sec^2 x$ but couldn't do justice. ...
0
votes
0answers
15 views

Principal value with truncation in $y$-direction

The Cauchy principle value uses truncation in $x$-direction, e.g$$PV\int_{-1}^1 \frac1x \, \mathrm{d}x = \lim_{\varepsilon \searrow 0} \int_{-1}^{-\varepsilon} \frac1x \, \mathrm{d}x + ...
2
votes
2answers
718 views

How to integrate $\sqrt{1-\sin 2x}$?

I want to solve the following integral without substitution: $$\int{\sqrt{1-\sin2x}} \space dx$$ I have: $$\int{\sqrt{1-\sin2x}} \space dx = \int{\sqrt{1-2\sin x\cos x}} \space dx = ...
1
vote
1answer
26 views

Let $(X,\mathcal{F},\mu)$ be a measure space and let $g\in L^1((X,\mathcal{F},\mu))$.

Let $\phi:[0,1]\to\mathbb{R}$ defined by $$\displaystyle \phi(t)=\int_X \frac{t^3g}{1+t^2g^2}\ \mathsf d\mu$$ Show that $\operatorname{Im}(\phi)\subset\mathbb{R}$ and that $\phi$ is continuous. ...
2
votes
1answer
32 views

How to integrate: $\int \frac{\sec x}{\sqrt{\sin(2x + A) + \sin A}} dx$?

How do I integrate: $$\int \frac{\sec x}{\sqrt{\sin(2x + A) + \sin A}}\, dx?$$ First, I tried to substitute $t^2$ for the denominator, but it was really a great flop. I then removed $\sin A$ since ...
0
votes
1answer
11 views

Finding the mass of a curve having a specified linear density using a line integral

I have some doubts whether the result I obtained is correct. As the topic title says - I am looking for the mass of a curve with a density of $$\sigma(x,y)= \sqrt{x}$$ The curve K is described as ...
8
votes
9answers
11k views

Evaluate $\int \frac{1}{\sin x\cos x} dx $

Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $ I know that the correct answer can be obtained by doing: $\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos ...
0
votes
0answers
14 views

Query about estimating an integral in Heat Equation

While studying the Heat Equation (P-309) from the book : 'Front Tracking From Conservation Laws' by Holden & Risebro; I have gone through the following calculation: " $\int_{\mathbb R} ...