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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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
14 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 .
-3
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
vote
3answers
39 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
57 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
187 views
4
votes
3answers
142 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
20 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
135 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 $$ g=(f')^{-1}? $$
1
vote
2answers
49 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
104 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 ...
2
votes
2answers
39 views

One integral involving integrals exponential and logarithmic function

Is there a closed-form solution for the integral $$ \int_{0}^{\infty}\log_{2}(1+ax)\cdot e^{-bx} \; \mathrm dx $$ with $a, b \geq 0$? If there is no closed-form solution, whether there is an ...
2
votes
0answers
35 views

How to solve this equation with implicit sum

I want to know how the authors of this arxiv paper (p. 10) solved the equation \begin{align} g\left(\lambda\right) ={}& ...
3
votes
0answers
58 views

A integral inequality

Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\left( ...
0
votes
1answer
88 views

What is the Indefinite Integral of $f(x)=x^x$? [duplicate]

A very simple question, but I cannot seem to find the answer. I know that $f(x)=x^x$ simply isn't integrable, but that doesn't mean the indefinite integral doesn't exist, right? Is there anything that ...
2
votes
1answer
84 views

My complex integral cancels at the end; how can I modify the integrand to prevent this?

$$\int_0^\infty \frac{x^a}{x^2 + b^2}$$ for $-1< a < 1$ and b>0 -- these constraints help with estimating the integral on the big circle and small circle of a keyhole contour that I chose to ...
2
votes
1answer
52 views

If $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$

Show that if $\mu(f>0)<1$ then $\lim\limits_{p\to 0^+}||f||_p=0$ Hint: Use Hölder's inequality. But I can't see where I should use it. I'm trying to use it in $\displaystyle\int |f|^p\,d\mu = ...
4
votes
5answers
142 views

$1+2+3=\int_{0}^{\infty}t^3e^{-t} dt$?

I'm reading Ivanov's: Easy as Pi. In the cover of the book, there is a formula: $$1+2+3=\int_{0}^{\infty}t^3e^{-t} dt$$ It's not clear to me if the formula has any relevance or if it is a joke. I ...
2
votes
3answers
258 views

One Step Forward from Gaussian Integral

Now to solve the integral $ \int_0^\infty e^{-x^2} \, dx $ has become a simple task for us. But how can we solve this integral: $$\int_0^\infty e^{-x^3} \, dx $$
3
votes
1answer
37 views

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

-> The sequence $(\int|f_n|^p\,d\mu)_{n \in \Bbb N}$ is bounded. -> $f_n\to f$ in measure. Prove that f is p-integrable. I'm trying to use the dominated convergence theorem. But I can't find an ...
3
votes
4answers
139 views

Can I use an upper semi-circle to integrate this function?

I'm trying to integrate $$\int_{-\infty}^{\infty} \frac{e^{iz}}{e^z + e^{-z}}dz$$ Do I have have to integrate this over a box, or can I use my first guess at a contour and use an upper semi-circle ...
-1
votes
1answer
67 views

How can I prove $x^3\, \frac{d^3 y}{dx^3} = \Delta(\Delta-1)(\Delta-2)y$?

This equation is used to solve Cauchy Euler Equation As it can be seen author has provided explanation of the fact how ...
2
votes
2answers
54 views

Apparent discrepancy between change of variables in one versus multiple dimensions.

My freshman calculus book gives the change of variables formula in one dimension and then eight chapters later gives it in $n$ dimensions. But when it generalizes to $n$ dimensions it requires the ...
1
vote
1answer
69 views

Show that $\int_0^\infty \frac{\sin(x)}{x}e^{-xt}\,\mathrm{d}x=\frac{\pi}{2}-\arctan(t)$; $t>0$

I did this Let $I=\int_0^\infty\frac{\sin(x)}{x}e^{-xt} \,dx$ Then, $\frac{\partial I}{\partial t}=\frac{\partial}{\partial t} ...
1
vote
4answers
56 views

Integrating for a solution in terms of an natural logarithm

Evaluating the following integral: $$\int_1^2 \frac2{1-3x}\ dx$$ why do you have to take the factor of $-2/3$ out when evaluating the integral?
2
votes
0answers
50 views

Approximate an integral with Bessel functions

Given $r,a,\lambda\in\mathbb{R}$, $r<a$, how can I find an approximate solution for the following definite integral? $$ \int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 ...
4
votes
1answer
47 views

Does $\lim\limits_{n \to +\infty} \left(\frac{n}{f(1)} \int_0^1 x^n f(x) dx \right)^n$ exists?

My question is having following one as a root. On one side, for $f : [0,1] \to \mathbb R$ continuous, one can prove that $$\lim\limits_{n \to +\infty} n \int_0^1 x^n f(x) dx =f(1)$$ On the other ...
6
votes
5answers
143 views

$\int_{0}^{\infty}\frac{dx}{a^2+\left(x-\frac{1}{x}\right)^2}$ equals $\frac{\pi}{5050}$

For $a\geq2$,if the value of the definite integral $\int_{0}^{\infty}\frac{dx}{a^2+\left(x-\frac{1}{x}\right)^2}$ equals $\frac{\pi}{5050}$.Find the value of $a$. Substituting $x-\frac{1}{x}=t$ does ...
1
vote
1answer
91 views

Show that the integral can not exceed $\frac{\pi^2}{96}$

Show that $$ \int_{-\infty}^{\infty}\int_{-x}^{x}\int_{-y}^{y}\int_{-z}^{z}e^{-(x^2+y^2+z^2+w^2)}\dfrac{|zw|}{(1+x^2)(1+y^2)} \,dw \,dz\,dy\,dx\le\frac{\pi^2}{96} $$ I am not understanding how $\pi$ ...
1
vote
4answers
126 views

Find $\int\frac{x-1}{x^2-5x+6}dx$. Why my solution is different from book?

I'm learning single variable calculus right now. Right now trying to understand integration with partial fraction. I'm confused in a problem from sometime. I think I'm doing right but answer in my ...
1
vote
2answers
41 views

Another question on bounded function , infinite partition and integrability

I am looking for an example of a bounded real valued function $f$ on some closed bounded interval $[a,b]$ such that for some infinite partition $\{a=c_0 , c_,c_2 , ...\}$ of $[a,b]$ , $f$ is Riemann ...
-1
votes
2answers
45 views

Evaluation on a basis of gaussian integral

Knowing that $$\int_{-\infty}^\infty e^{-x^2} dx= \pi^{\frac{1}{2}}$$ Find: $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{\frac{-x^2}{2}} dx$$ And my question is how does this help if have the ...
1
vote
1answer
38 views

Boundaries change in double integral

Calculate: $$\int_0^1 \int_0^{x^3} e^\frac{y}{x} dydx$$ Obviously i need to change it to $dxdy$ thus i need to change the boundaries of the second integral but how to do that in this case?
1
vote
2answers
23 views

Double integral on a compact subset

Calculate: $$\int \int _D \left(6x+2y^2 \right) dxdy$$ where D is a compact subset of $\mathbb{R}^2$ enclosed by a parabola $y=x^2$ and a line $x+y=2$. How to find that, how to find the limits of ...
2
votes
0answers
40 views

How can one conclude that $f$ is differentiable?

How can one conclude that $f$ is differentiable ? If $f:\mathbb R\to S^1=\{z\in\mathbb C:|z|=1\} $ is continuous and such that $f(a+b)=f(a)f(b)$ then the formula holds $\displaystyle ...
1
vote
0answers
15 views

What happens to tangential gradient when flattening a surface

The tangential gradient $\nabla_\tau f$ associated to a surface $S$ is defined as the projection of a suitable extension $\nabla f$ to the tangent plane to that surface. It seems reasonable to think ...
2
votes
3answers
289 views

Double integral problem: $\int_0^\pi\int_x^\pi \frac{\sin y}{y} dy\, dx$

Calculate: $$\int_0^\pi \int_x^\pi \frac{\sin y}{y} dydx$$ How to calculate that? This x is terribly confusing for me. I do not know how to deal with it properly.
0
votes
0answers
59 views

What's wrong with my integration by parts?

I'm trying to calculate an integral by using integration by parts of the following integral: $\pm ...
0
votes
1answer
58 views

Differential Equations first order [closed]

Anyone who can help me on this equation, $y' = (\frac{y}{x + y^3})$ I've already tried to make a substitution which is: $h(x,y) = x+y^{3}$ and did the derivatives but still no solution, so if anyone ...
1
vote
1answer
50 views

Find centre of mass of a circle when one half is heavier than the other half?

I have a problem which simply states: Consider a circle (lamina) of radius 1 with centre (0,0) where the left half is twice as heavy as the right. Find its centre of mass. Extend your solution to ...
3
votes
4answers
91 views

Find $\int_{1}^{2}\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx$

$$\int_{1}^{2}\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx=\frac{u}{v}$$ where $u$ and $v$ are in their lowest form. Find the value of $\dfrac{1000u}{v}$ ...
0
votes
4answers
64 views

Find $\int_{3}^{5}\left(\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\right)dx$

Evaluate the integral $$\int_{3}^{5}\left(\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\right)dx$$ I tried replacing $x$ by $x-5-3$ but this is not working,neither Simpsons approximation is working.Is ...
0
votes
1answer
46 views

Find $\int_{a}^{b}\frac{x^{n-1}\left((n-2)x^2+(n-1)(a+b)x+nab\right)}{(x+a)^2(x+b)^2}dx$

Prove that $$\int_{a}^{b}\frac{x^{n-1}\left((n-2)x^2+(n-1)(a+b)x+nab\right)}{(x+a)^2(x+b)^2}dx=\frac{b^{n-1}-a^{n-1}}{2(a+b)}$$ I tried replacing $x$ by $x-a-b$ but problem is getting messy and ...
3
votes
1answer
106 views

Prove that $\int_0^{a}{\int_x^a{t^{-1}f(t)dt}} = \int_0^a{f(x)dx}$ [duplicate]

I got stuck on this problem from Real Analysis by Folland. Can anybody give me any hints on how to solve this? If $f$ is Lebesgue integrable on $(0, a)$ and $$ g(x) = \int_x^a{t^{-1}f(t)dt} $$ ...
0
votes
0answers
18 views

Deforming path of integration from the real line to the boundary of a open subset of the upper half complex plane.

Denoted the upper half of the complex plane by $\mathbb{C}^{+}=\{z\in\mathbb{C}:\text{Im }z>0\}$. Let the open, unbounded set $A\subseteq\mathbb{C}^{+}$ have a boundary $\partial A$ such that the ...