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Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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Parameterising Surfaces Integration

$S$ is the surface of a cube which is bounded by $6$ planes being $x=1,x=3,y=2,y=4,z=0,z=2$. The normal vector points outwards and the vector field is $F = (x^2-sin(yz),\frac{cos(x)}{x^2}-yz,x^2y)$ ...
Dam's user avatar
  • 259
1 vote
1 answer
42 views

Find out $\int_{0}^{\infty} \frac{x^2 + ax + 1}{1 + x^4} \arctan\left(\frac{1}{x}\right)\rm{d}x$ .

Find out $$\int_{0}^{\infty} \frac{x^2 + ax + 1}{1 + x^4} \arctan\left(\frac{1}{x}\right)\rm{d}x$$ My attempt: $$I = \int_{0}^{\infty} \frac{x^2 + ax + 1}{1 + x^4} \arctan(\frac{1}{x})\rm{d}x = \int_{...
Ash_Blanc's user avatar
  • 749
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0 answers
20 views

Justification of the change of path of integration of $\Gamma$ in this article [duplicate]

I was reading an article and the author wrote \begin{equation*} \int_{z = -2 \pi i \alpha n \mathbb{R}_+}e^{-z}z^{s-1}dz = \int_{z = \mathbb{R}_+}e^{-z}z^{s-1}dz \end{equation*} where $s= 1/2 +...
Gaelink's user avatar
  • 31
2 votes
0 answers
36 views

Does the integral converge $\int\limits_{0}^{1}\frac{\cos\left ( t^{-2} \right )}{\left ( 2-t^2\cos\left ( t^{-2} \right ) \right )^2}t^{a}dt$?

Investigate the convergence and absolute convergence of the integral at $a\in \mathbb{R}$ $$I=\int\limits_{0}^{1}\frac{\cos\left ( t^{-2} \right )}{\left ( 2-t^2\cos\left ( t^{-2} \right ) \right )^2}...
Dmitry's user avatar
  • 1,382
1 vote
0 answers
12 views

Brownian Motion $\mathbb{P}\left[W_t \geq x\right]$ integrating by parts

Prove that if $\left\{W_t\right\}_{r \geq 0}$ is standard Brownian motion under $\mathbb{P}$ then, for $x>0$, $$ \mathbb{P}\left[W_t \geq x\right] \equiv \int_x^{\infty} \frac{1}{\sqrt{2 \pi t}} e^{...
Allison's user avatar
  • 205
-5 votes
2 answers
64 views

Integrate $ \int_{} \frac{\sin(x-a)}{\sin(x-b)} {\rm{d}}x$ smartly than rest?

Integrate $ \int_{} \frac{\sin(x-a)}{\sin(x-b)} {\rm{d}}x$ I am expecting better new elegant approaches for this simple integral and lets see the no. of other ways to solve it. My try : $$I = \int_{}...
Ash_Blanc's user avatar
  • 749
3 votes
2 answers
53 views

Calculate $I(\alpha)=\int_0^\pi \frac{x}{1+\sin(x)\cos(\alpha)}dx$

I have tried to derive it 1 and 2 times but I do not see any relation to the original $I(\alpha)$, is there any breakthrough that I am missing?
MiguelCG's user avatar
  • 307
1 vote
1 answer
47 views

Integrate $ \int_{} 1 + \tan x\tan(x+\theta) {\rm{d}}x$ quickly .

Integrate $ \int_{} 1 + \tan x\tan(x+\theta) {\rm{d}}x$ I am expecting better new elegant approaches for this simple integral and let us see how shorter it can be. My try : $$I = \int_{} 1 + \tan x\...
Ash_Blanc's user avatar
  • 749
-1 votes
0 answers
37 views

I cannot crack the following integral, can you help me out?

I've been reading up on the following paper (plugging it in here for reference https://arxiv.org/pdf/2312.06878) where I'm trying to reproduce some of the results they found, in particular eq 3.19. ...
Sophie Schot's user avatar
3 votes
2 answers
65 views

Integrating $\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$ two ways gives very different-looking answers. How to show algebraically they differ by a constant?

From Apostol's Calculus Volume 1 2nd ed., 6.22 #46, the task is to integrate $$\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$$ Method 1: The provided hint is to use the substitution $x-a=(b-a)\sin^2(u)$. Thus $b-...
newmacuser's user avatar
2 votes
1 answer
91 views

Proper way of computing $\int_{0}^{\infty}\frac{\ln(\cos^2(x))}{x^2}dx$??

I noticed that people solved the following integral using Lobachevsky trick. However, notice that in $$\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}\cdot\frac{\ln(\cos^2(x))}{\sin^2(x)}dx$$ the function $\...
Silver's user avatar
  • 767
4 votes
2 answers
97 views

The "seashell constant": closed form for $\frac12\exp\int_0^1-\log(\sin(\frac{\pi}{6}+\frac{2\pi}{3}x))\mathrm dx$?

I am looking for a closed form for $R=\frac12\exp\int_0^1-\log\left(\sin\left(\frac{\pi}{6}+\frac{2\pi}{3}x\right)\right)\mathrm dx\approx0.6159$. Wolfram does not give a closed form for $R$. Wolfram ...
Dan's user avatar
  • 23.7k
1 vote
0 answers
93 views

What is the Name of this Integration Trick?

There is this trick I've learned for solving integrals but I'm unsure if this technique has a certain name. I'll give two examples below. $$\int\frac{x}{x+1}dx$$ $$\int\frac{x+1-1}{x+1}dx = \int\left(...
Grey's user avatar
  • 308
1 vote
2 answers
51 views

Evaluate $\int_{0}^1x^n\ln(x)^mdx$ [duplicate]

$$ \mbox{I have to evaluate the integral}\quad \int_{0}^{1}x^{n}\ln^{m}\left(x\right){\rm d}x $$ where I do not have much information about $n,m$, so I suppose they are natural numbers. In that case, ...
MiguelCG's user avatar
  • 307
2 votes
0 answers
34 views

Compute integral $\int_{\partial B(0,r)} \frac{\log|z-y|}{|z-x|^2}\,dz$

I would like to compute the following integral (in two dimension): $$ \int_{\partial B\left(0,r\right)} \frac{\log\left(\left\vert z - y\right\vert\right)} {\left\vert z - x\right\vert^{2}}\,{\rm d}z\...
kzzz's user avatar
  • 99
10 votes
1 answer
667 views

Integral's computation does not match WolframAlpha result

So I'm computing the following integral: \begin{align} \int_0^1 \frac{\tan^{-1}(x)\ln(x)}{x}dx \end{align} I started out with a simple integration by parts which yielded: \begin{align} -\int_0^1 \frac{...
NEON's user avatar
  • 148
4 votes
1 answer
94 views

How can I show that $\int_0^{\frac{\pi}{2}}\sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right) \ dx$.

Question: How can I show that $$\int_0^{\frac{\pi}{2}} \sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right) \ dx=\log\left(\left(2\sqrt{2-\sqrt{2}}+2\sqrt{2}-1\right)^{\sqrt{2+\sqrt{2}}} \...
Martin.s's user avatar
  • 4,254
3 votes
2 answers
85 views

Evaluate the integral $\int_0^\infty \frac{\ln(1+u)}{u+u^\phi} \mathrm du$

$$\int_0^\infty \frac{\ln(1+u)}{u+u^\phi}\mathrm du$$ I saw this problem posted before, however it was deleted. I forgot the solution to this integral but I don't know how to begin. Maybe I should ...
uggupuggu's user avatar
  • 427
0 votes
0 answers
48 views

Manual integration: Proving the Laplace transform: $\frac{s-a}{(s-a)^{2}+k^{2}}$ via $e^{4t}\cos(-6t)$

Last spring I was searching information for solving Laplace transforms. There was plenty of information in my textbook and on sites like MIT-open Courseware, YouTube, etc. However, I was asked to ...
Mitchell's user avatar
0 votes
0 answers
35 views

Let f : [0, a] → R be given by f(x) = x 2 . Find R a 0 f(x)dx [closed]

Let f : [0, a] → R be given by f(x) = x 2 . Find R a 0 f(x)dx
Vidushi Agarwal's user avatar
-2 votes
1 answer
23 views

Problems understanding integration in a banach fixed point exercise [closed]

I have been studying for a math test and I have encountered the following problem: banach's fixed point theorem applied to finding a unique function I do not get the last step. I do not know how ...
valentina manfredi's user avatar
-2 votes
0 answers
32 views

How to solve $\int_{-i}^i \frac{1}{x} dx$, and questions about complex integration. [closed]

last night I stumbled upon this problem at about 3 am due to my own random searching in math. [My math level is I did calc 1 and 2 and I am an engineering student who just finished first year] My ...
William Banquier's user avatar
6 votes
1 answer
77 views

$\int\limits_{0}^{\pi}{\frac{\cos^2{\left(\frac{\pi}{2}\cdot\cos(x)\right)}}{\sin(x)}\;\mathrm{d}x}$

I am trying to solve this integral: $$\int_{0}^{\pi}{\frac{\cos^2{\left(\frac{\pi}{2}\cdot\cos(x)\right)}}{\sin(x)}\;\mathrm{d}x}$$ I know that it must have something to do with the Cin function: $$\...
FixFox's user avatar
  • 61
1 vote
1 answer
33 views

Solution to a First-Order PDE Using the Lagrange method and division by zero

I need to find a solution for the following PDE: $$ \begin{cases} y u_x - x u_y = 0 \\ u(x, 0) = x^2 & ,x \in \mathbb{R} \\ \end{cases} $$ My attempt: I started by using the ...
knight5478's user avatar
0 votes
0 answers
29 views

Integral of test functions

Assume that $\phi$ is a test function then for $t \in \left(0,T\right)$ and $x \in \Omega$ where $\Omega$ is an $open,\ connected\ set\ \mbox{with}\ smooth\ boundary$. Is the following claim true ?: $$...
user1174736's user avatar
0 votes
0 answers
31 views

How to integrate this integration very similar to the Fresnel integral?

Thank you for reading my question! I want this integral $$ \int_0^x\sin{(\alpha+t^2)}dt $$ I know the Fresnel integral is $$ S(x)=\int_0^x\sin{(t^2)}dt\\ C(x)=\int_0^x\cos{(t^2)}dt $$ And we can ...
Xiangyu Cui's user avatar
3 votes
2 answers
78 views

$\int\frac{\sin 2x\sin 3x}{\sin 2x+\sin 3x}\,\mathrm dx$

$$\int\frac{\sin 2x\sin 3x}{\sin 2x+\sin 3x}\,\mathrm dx$$ My attempt: Rewriting the numerator: We can use the double angle identity for sine: $$\sin(A)\sin(B) = \frac12\left(\cos(A - B) - \cos(A + B)\...
Nsnansn Jwjwj's user avatar
-1 votes
0 answers
102 views

$f(x) \ge 0$ for all $x \in [a, b]$ and $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$.

Suppose that $f$ is continuous on $[a, b]$, that $f(x) \ge 0$ for all $x \in [a, b]$ and that $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$. My attempt: Let $\dot{\Pi}$ be a tagged ...
user13's user avatar
  • 1,669
1 vote
1 answer
31 views

Changing the integration limits of a triple integral

I have a triple integral of the form $$ \int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_1} dt_3\ f(t_1,t_2,t_3) $$ and I want to transform it to the form $$ \int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_2} dt_3\...
SrJaimito's user avatar
  • 451
0 votes
1 answer
38 views

Derivation of the volume of a half-sphere

I am trying to derive the volume of a half-sphere with a constant r using integration. Integration I first try to integrate 90 degrees from the top of the halfsphere down to the bottom plane of the ...
Jan F. S's user avatar
3 votes
6 answers
121 views

What is $ \int_{0}^{\frac{1}{\sqrt{3}}} \sqrt{x + \sqrt{x^2 + 1}} \rm{d}x$?

What is $ \int_{0}^{\frac{1}{\sqrt{3}}} \sqrt{x + \sqrt{x^2 + 1}} \rm{d}x$ ? I m not getting any idea for this one . Expecting from the community to share some interesting substitution ideas / tricks ...
Ash_Blanc's user avatar
  • 749
5 votes
1 answer
60 views

Solving the integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \ dx$

My textbook has the following problem: $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \ dx$$ Using the trigonometric identity: $$\sin 2x = 2 \sin x \cos x$$ the ...
archthegreat's user avatar
2 votes
1 answer
28 views

$\mathscr{L}^{p_1}(\mathbb{N},\mathscr{A},\mu)\subseteq\mathscr{L}^{p_2}(\mathbb{N},\mathscr{A},\mu)$: $1\leq p_1<p_2<+\infty$, $\mu$ counting measure

I need to prove the following: Suppose that $1\leq p_1<p_2<+\infty$. Let $\mu$ is the counting measure on the $\sigma$-algebra $\mathscr{A}$ of all subsets of $\mathbb{N}$. Then $\mathscr{L}^{...
Beerus's user avatar
  • 1,595
2 votes
1 answer
52 views

Integration contour of $\int_0^L\frac{e^{-ikx}\sin\left(nx\right)}{\sin(x)}dx$?

As the title says, I am trying to solve the Fourier integral \begin{align} I=\int_0^L\frac{ e^{-ikx}\sin\left(n x\right)}{\sin x}dx, \end{align} where $n\in\mathbb{Z}$ and $L$ if finite. I have ...
hyriusen's user avatar
  • 147
0 votes
0 answers
36 views

Using the Maclaurin series of sin(x) to find Si(x)

The Maclaurin series for $\sin\left(x\right)$ is given by: $$\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}x^{2n+1}}{\left(2n+1\right)!}$$ So the series for $\frac{\sin\left(x\right)}{x}$ would be $$\...
Gabriel Turner's user avatar
1 vote
1 answer
54 views

How to integrate $\int \frac{x^{2}+7(\sqrt[m]{x^{m}})+6}{\sqrt[m]{x^{m}}+1}dx$?

Q)How to integrate $\int \frac{x^{2}+7(\sqrt[m]{x^{m}})+6}{\sqrt[m]{x^{m}}+1}dx$ ? Here, $m=2n$ and $n=$ natural number. First of all my approach: Let $$I=\int \frac{x^{2}+7(\sqrt[m]{x^{m}})+6}{\sqrt[...
Dropper's user avatar
  • 77
2 votes
1 answer
49 views

Limit of sequence defined by an integral

Given a sequence $I_n = \int_{2\pi n}^{2\pi(n+1)}\frac{\sin x}{x}dx$, prove the sequence is monotone and find the limit $\lim _{n\rightarrow \infty}I_n$. How should I tackle these problems? It is not ...
anonymous_wolf's user avatar
6 votes
2 answers
90 views

How to Evaluate the Integral $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$

Question: How to Evaluate the Integral $$\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$$ My attempt I'm looking for a method to evaluate it. I've attempted a substitution to ...
Martin.s's user avatar
  • 4,254
2 votes
3 answers
97 views

Cheap way of estimating $\int_0^{+\infty}\frac{\sin^4x}{x^4}\operatorname dx$

Let $I=\int_0^{+\infty}\frac{\sin^4x}{x^4}\operatorname dx$, find the integer part of $10I$. In fact, it is possible to show that $I=\frac\pi3$. But I think the process is too technical. Since the ...
youthdoo's user avatar
  • 1,291
1 vote
3 answers
80 views

Solving the integral $\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\ dx}{\cos^2 x + 4\sin^2 x}$ [duplicate]

My textbook has the following problem: $\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\ dx}{\cos^2 x + 4\sin^2 x}$. This website solves the problem in a very convoluted way and I was wondering if there's an ...
archthegreat's user avatar
3 votes
1 answer
134 views

What is $\int_{0}^{\infty} \frac {\ln x}{{({1-x+x^2})}^2} \rm{d}x$?

What is $\int_{0}^{\infty} \frac {\ln x}{{({1-x+x^2})}^2} \rm{d}x$ ? I m not getting any idea for this one . Expecting from the community to share some interesting substitution ideas / tricks . My ...
Ash_Blanc's user avatar
  • 749
-1 votes
1 answer
72 views

What is $\int_0^{\infty}\sin(ax)\cos(bx)dx$ ? where $a$ and $b$ are constant. [closed]

I am solving an E&M problem and I can't get the solution of integration above. I guess it is something to do with residue theorem, but I still can't solve it. Please help...
KingWangZZang's user avatar
4 votes
1 answer
95 views

how to evaluate $\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{\sqrt{xy}(1+xy) \log_{\pi}^{2}{xy}} \, dx \, dy$

How to evaluate: \begin{align*} &\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{(\sqrt{xy}+\sqrt{x^{3} y^{3}}) \left[ \log_{\pi}^{2}{x} + 4\log_{\pi}{\sqrt{x}} \log_{\pi}{y} + \log_{\pi}^{2}{y} \right]} \, ...
Martin.s's user avatar
  • 4,254
0 votes
0 answers
49 views

Leibniz Integral Rule with Ito Integral

From a text on mathematical finance and stochastic calculus, I am trying to understand how Leibniz integral rule is being applied when the integrator is dWt (a standard Wiener process): Later the ...
dwcecil's user avatar
  • 125
1 vote
1 answer
55 views

Study if the function $F(x)=\int_{0}^1D(xu)du$ is differentiable

Given the function $D(x)=\inf\{|x-n|:n\in \mathbb{N}\}$, study the differentiability of $F(x)=\int_{0}^1D(xu)du$. My try in $0$: $\lim_{h\to 0}\frac{F(h)-F(0)}{h}=\lim_{h\to 0}\frac{\int_{0}^1D(hu)du}...
MiguelCG's user avatar
  • 307
1 vote
0 answers
26 views

Expected value of mean reverting spot price - triple numerical integral

I have been reading this paper: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=7d38b03cfc62a15bdfd755c793d4e70a821725cc and having trouble trying to implement the expected ...
loprocto's user avatar
-1 votes
0 answers
60 views

Proof Integral inequality [duplicate]

Is it true that for any continuous function $f(x)$ on $[0,1]$ the next inequality $$ \int_0^1\int_0^1|f(x)+f(y)|dxdy \geq \int_0^1|f(x)|dx$$
Vertum's user avatar
  • 331
2 votes
0 answers
97 views

What's the value of $\int \frac{f''(x)}{f(x)} dx$?

Motivation: This was a printing error in my MCQ test, correct question contained the first order derivative. But it's destroying my mental peace. My attempt I don't even know where to start. This ...
Gwen's user avatar
  • 2,544
5 votes
3 answers
168 views

Is it possible to integrate $\int \frac{x^2 - 2x + 3}{x^4 - x^3 + x^2 - x + 1} \ dx$ using methods typical of undergraduate calculus?

I was doing a calculus problem and got to this final integral: $$\int\limits_{0}^{1} \frac{x^2 - 2x + 3}{x^4 - x^3 + x^2 - x + 1} \ dx.$$ Calculators are supposed to be used for these problems, so I ...
ten_to_tenth's user avatar
4 votes
0 answers
72 views

When is $ \sum_{k \in \mathbb{Z}}\left(\frac{\sin(k)}{k}\right)^{n}=2 \int_{0}^{\infty}\left(\frac{\sin(x)}{x}\right)^{n}dx$?

Define sequences $$a_n = \sum_{k \in \mathbb{Z}}\left(\dfrac{\sin(k)}{k}\right)^{n}, b_n = \int_{0}^{\infty}\left(\dfrac{\sin(x)}{x}\right)^{n}dx.$$ I am trying to see if there is a relation between ...
Sam's user avatar
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