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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0
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1answer
65 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
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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 ...
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 ...
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 ...
4
votes
2answers
89 views

Double integral with a product of dilog $\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x)\ dx \ dy$

One of the integrals I came across these days (during my studies) is $$\int _0^1\int _0^1\text{Li}_2(x y) \text{Li}_2((1-y) x) \ dx \ dy$$ that can be turned into a series, or can be approached by ...
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 ...
1
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0answers
31 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
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?
-1
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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 ...
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
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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
0
votes
2answers
51 views

Integrate the following problems

Let us consider the integral $$\int \frac{x^2}{(x\sin x+\cos x)^2}\, dx$$ I have tried with the following way $\displaystyle \int \frac{x^2}{(x\sin x+\cos x)^2}\, dx$ $\displaystyle \Rightarrow ...
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} ...
4
votes
2answers
58 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 ...
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 ...
18
votes
4answers
189 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
87 views

$\int_{0}^{1}\frac{x^{n-1}}{1+x^n} \log\log{(\frac{1}{x})}dx =-\frac{\log(2)\log(2n^2)}{2n}$ [on hold]

Does anyone prove the following definite integral ? $$\int_{0}^{1}\frac{x^{n-1}}{1+x^n} \log\log{(\frac{1}{x})}dx =-\frac{\log2(\log2+2\log(n))}{2n}$$
0
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0answers
55 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 ...
1
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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
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 ...
5
votes
4answers
101 views

Limit of definite integral of $f(x)\cos(mx)$

Source: Old comp./preliminary exam. Let $f(x)$ be a Riemann integrable function on $[0,1]$. Prove that $$\lim_{m\to\infty}\int_{0}^{1}f(x)\cos(mx) \, \,dx=0$$ Thought $(1)$ Because we don't know if ...
0
votes
0answers
14 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
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 ...
1
vote
1answer
25 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. ...
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 ...
0
votes
4answers
81 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
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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} ...
3
votes
3answers
137 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.
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 ...
2
votes
1answer
72 views

Generalization to this integral

$$ \int_0^\infty \frac{\ln(1 + x^a)x^s}{1+x^2} \ dx $$ Actually the problem was $ \displaystyle \int_0^\infty \frac{\ln(1 + x^a)}{(1+x^2)\ln(x)} \ dx $. But I guess the form of a Mellin Transform ...
2
votes
1answer
52 views

Can I integrate an approximate equality?

I have a function $f(x)$ and its first derivative, which is continuous, $f'(x)$. I know that $\lim_{x\to\infty}f'(x)=0$. Also $f'(x)>0$ for all $x$. I also have another function $p(x)$ which is a ...
0
votes
0answers
13 views

Integrate in cylindrical coordinates

$\vec{\nabla} p = \rho \vec{f}$ How do you solve this in polar coordinates? I can't find a way to insert g in my equation. I would have to split it in $r$ and $\phi$. So $y=sin(\phi) * r$ but I ...
0
votes
3answers
74 views

How to integrate : $\int \sqrt{\tan^2x +2}dx$ [closed]

How to integrate : $\int \sqrt{\tan^2x +2}dx$ Please guide what to substitute or any approach as I am not getting any clue on this , thanks .
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votes
2answers
51 views

How to integrate $\int_{0}^{1} \frac{1 - ( 1 - x )^n}{x} \, dx$? [closed]

How to integrate $$\int_{0}^{1} \frac{1 - (1 - x)^n}{x} \, dx \ ?$$
1
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3answers
38 views

How to integrate greatest integer function x from -1 to 2? [closed]

Can some one please help me to solve $\int_{-1}^{2}\left \lfloor x \right \rfloor dx$ ?
1
vote
2answers
56 views

Why can I use Fubini' theorem on this function?

I used the fact that $\displaystyle \int_0^\infty\int_0^1 e^{-y}\sin(2xy)\,dxdy=\int_0^1\int_0^\infty e^{-y}\sin(2xy)\,dydx$ to solve $\displaystyle\int_0^\infty e^{-y}\frac{\sin^2(y)}{y}\,dy$. (The ...
6
votes
2answers
184 views
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
19 views

Efficient approximation to integration of analytic expressions involving product of four bessel functions

I have to take many integrals of the form $$ \int_0^\infty \!dx\,\,e^{-x}\,x^{\gamma - 2\beta - 2\alpha} j_\alpha ( u_1 x)j_\alpha (u_2 x)j_{\beta}(u_3 x)j_\beta (u_4 x),$$ where $\gamma$ is an ...
6
votes
4answers
134 views

Integrating $\int \sqrt{x+\sqrt{x^2+1}}\,\mathrm{d}x$

Integrating $$\int \sqrt{x+\sqrt{x^2+1}}\,\mathrm{d}x$$ Using substitution of $x=\tan \theta$, I got the required answer. But is there a more elegant solution to the problem?
2
votes
1answer
35 views

How to correctly write definite time integration of this function?

Last time I saw an integral was something like 10 years ago, and I am having doubts about the notation I should use. I want to describe the evolution of the volume difference between two cylinders ...
0
votes
1answer
32 views

How can $ (D^2 +1)y $ be solved such that it's equal to $x \cos x$?

Can anyone provide solution for $(D^2 +1)y$ such that it's equal to $ x \cos x$ or vice versa?
2
votes
1answer
53 views

Computing the integral of $-1/f''$

I think this is a very silly question but I have some problems nonetheless. If I know that $g'=-\frac{1}{f''}$, is then $$ g=(f')^{-1}? $$
-4
votes
0answers
51 views

I need some help. Find $\int \frac{\sin^2 x \cos x}{\sin x+\cos x}$ [closed]

$$\int \frac{\sin^2 x \cos x}{\sin x + \cos x}\mathrm dx$$
1
vote
2answers
48 views

Integrating Square Roots Containing Multiple Trigonometric Functions and/or Numbers

When trying to calculate arc length of a curve I frequently come across problems that I do not know how to integrate, such as: $$ \int{\sqrt{16\cos^2{4\theta} + \sin^2{4\theta}} d\theta} $$ Which in ...
4
votes
1answer
103 views

Changing the values of an integrable function $f:[a,b] \to \mathbb R$ countably infinitely many points not a dense subset of $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a function Riemann integrable over $[a,b]$ . It is known that if we change its values at finitely many points of $[a,b]$ , then the changed function still remains ...