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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-1
votes
2answers
37 views

Integration problem with $e$ and $\ln$

Can anyone help me solve this? I tried with integration with parts, but without luck. The function to be integrated is $$\frac{e^{x+\ln x}}{x}$$
0
votes
3answers
104 views

A simple looking integration : $\left(\frac{x^3}{1+x^5}\right)$

One of my friends gave me this problem about a week back and since then, I have been toiling to get a solution to this problem, but I just get stuck at some step. Can someone please tell me the steps ...
2
votes
5answers
102 views

Mistake in evaluating $\int\dfrac{dx}{\ln(x)}$

Evaluate: $$I=\int\dfrac{dx}{\ln(x)}$$ My attempt: $$$$ $$I=\int \dfrac{x'}{\ln(x)} dx$$Integrating by Parts,$$\dfrac{x}{\ln(x)}-\int\dfrac{x}{(\ln(x))'}dx$$$$=\dfrac{x}{\ln(x)}-\int ...
1
vote
0answers
33 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
-3
votes
2answers
38 views

Crazy and difficult Limits and integration

This limit take from me much time to solve and finally I can't. So please help me to solve.. Find $L$ $ L =\displaystyle\lim_{x\to \infty} \frac{\displaystyle\int_{1}^{x} t^{t-1} ( t + tln (t) +1 ) ...
0
votes
1answer
72 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
2
votes
1answer
32 views

Reference for differentiation of an integral over variable ball

I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$ \frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y), $$ where $dS$ is the Lebesgue surface ...
0
votes
0answers
21 views

Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and approximate $R_n < \frac{1}{10000}$

I am tasked with the following: Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and use the power series to approximate $\displaystyle \int_{0}^{0.5}3e^{-x^2/2}$ with error ...
6
votes
1answer
45 views

$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I$ [duplicate]

Let $I:=[a,b]$ and let $g: I \to \Bbb R$ be continuous on $I$. Suppose that there exists $K > 0$ such that $$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I.$$ Then $g(x) = 0\ \ \forall x \in I $. ...
2
votes
5answers
77 views

if we have $(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$ then $f(x) =x \ \forall x\geq0$.

Let $f: [0, \infty) \to \Bbb R$ be continuous and $f(x) \neq 0 \forall x>0$. If we have $$(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$$ then $f(x) =x \ \forall x\geq0$. We have $(f(x))^2 = 2 ...
0
votes
1answer
11 views

$F(x) := (n- 1)x-\frac{ (n- 1)n}{2}$ for $x \in [n- 1, n), n \in \Bbb N$ using this result to evaluate $\int_a^b[x]dx.$

Let $F(x)$ be defined for $x \geq 0$ by $F(x) := (n- 1)x- (n- 1)n/2$ for $x \in [n- 1, n), n \in \Bbb N$. Show that $F$ is continuous and evaluate $F'(x)$ at points where this derivative exists and ...
6
votes
0answers
27 views

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
1
vote
0answers
21 views

On utilizing the Leibniz rule of integration on a non compact interval.

I am following some slides that you can find here. At slide $\approx$ 24 a problem arises, to find $$\DeclareMathOperator*{\argmin}{\arg\!\min} \argmin_{\hat{y} } -\int_{-\infty}^{\hat{y}} (y ...
3
votes
3answers
50 views

How to find: $\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$?

How to find: $$\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$$ I tried multiplying it all out but I just ended up in a real mess and I'm wondering if there is something I'm ...
2
votes
1answer
60 views

Show that there exist continuous functions $g,h:[0,1]\rightarrow \mathbb{R}$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a Riemann Integrable function. Let $\epsilon>0$. Show that there exist continuous functions $g,h:[0,1]\rightarrow \mathbb{R}$ such that $g(x)\leq f(x)\leq ...
3
votes
2answers
77 views

Evaluate the integral $\int \frac{x}{a+bx^3}\ dx$

How do I solve integral at this form $\displaystyle\int \frac{x}{a+bx^3}\ dx$ ? I have tried a lot of things, but it doesn't work. I also know that the solution isn't that easy.
1
vote
3answers
77 views

Integral Convergence $\sin{x}/x^{3/2}$

Does the following integral converge: $$\int_0^\infty{\frac{\sin x}{x^{3/2}}}dx$$ I have tried to integrate this by parts and arrived at: $$-x^{-3/2}\cos x -\int \frac 12{x^{-1/2}}\cos{x} dx $$ ...
0
votes
0answers
72 views

$\int_0^b \ln(\sin(ax))dx$ [duplicate]

Problem: Evaluate $$\int_0^b \ln(\sin(ax))dx$$ Unfortunately I have no idea as to how to proceed with finding a closed form for the above Integral. The $a$ in the integrand made me think of ...
1
vote
1answer
59 views

Find the area using double integral and polar coordinates.

I need to find the area using double integral and polar coordinates. $$y=3-x$$ $$y^2=4x$$ This is what i figured already: $${r\cos{\theta}+r\sin{\theta}} = 3$$ $$r=0, r=3, \theta=0, \theta=\pi/2$$ ...
5
votes
5answers
97 views

$\int\dfrac{dx}{x^2(x^4+1)^{3/4}}$ [duplicate]

Evaluate $$\large{\int\dfrac{dx}{x^2(x^4+1)^{3/4}}}$$ I thought of rewriting this as $$\large{\int\dfrac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$ and substituting ...
2
votes
1answer
38 views

Double integral - Convert to polar coordinates and find the integration limits by a given domain [closed]

I need help converting to polar coordinates and find the limits of the integrals by this given domain: $$\iint_{D}{} f(x,y)\, dx\, dy$$ $$D= \left\{ (x,y) \mid \dfrac {x^2}{a} \leq y\leq a, -a\leq 0 ...
4
votes
3answers
99 views

Integrating volume of a sphere with a cylinder “drilled” out of it

Unfortunately, I am stuck again on another integration problem. Famous last words, this should be simple. $$ \text{A cylindrical drill with radius 5 is used to bore a hole through}\\\text{the center ...
1
vote
2answers
36 views

partial fraction derivative question

So I have this partial fraction derivative question. I know how to solve it, but for some reason I keep swapping two numbers. Here is the problem: $$\int\frac{3-4x}{x^2+x}= ...
0
votes
1answer
25 views

Integrating a surface bound by a circle

I'm having an issue setting up this problem correctly, regardless of how I seem to do it I end up canceling everything out and getting $0$, which isn't the correct answer. $$ \text{Find the surface ...
1
vote
0answers
22 views

Converting cartesian to polar integral

I feel like I almost have a grasp on regions of integration, I am a bit frustrated that I haven't fully gotten it but because I feel like I'm almost there. In this particular homework problem I have a ...
-3
votes
2answers
36 views

How to find the area under a semicircle using integration? [closed]

How would I go about finding the area under a semicircle? I know that to use integration the formula is $\int_a^b f(x) \mathrm{d}x.$However, when I put this into my graphing calculator it doesn't ...
0
votes
2answers
16 views

Setup region of integration for polar coordinates

I've been working on a homework set for Calc III, right now we're emphasizing double integration and polar integrals. I keep having problems conceptualizing where to actually create my region of ...
1
vote
2answers
32 views

Find total area under infinite curves

My question is finding the total area covered by curves, such as the total area every curve in the following picture covers (from 100 on y axis to 200 on x axis): In my case, the curves are ...
1
vote
1answer
68 views

Stuck on integration question

The curve in the picture shown has equation $y=bx(x-2)$ (a) Find b given that the shaded area is 4 units$^2$ (b) Find the x-coordinate of the point A if the line OA divides the shaded area into ...
0
votes
0answers
25 views

Can you prove that the integral below, with a vectorial field, is zero?

If $\vec{J}(\vec{r})$ is a vector field limited in infinity. Prove that the integral below is zero: \begin{equation} ...
2
votes
0answers
56 views

How to integrate the following sum?

I'm currently trying to show: $$ \int_0^1{\int_0^y{\sum_{n=0}^{\infty}\left(\frac{1}{10^{n+1}x(1-x)}\left(9+\frac{1}{1-x^{10^n}}-\frac{10}{1-x^{10^{n+1}}}\right)\right)dx}dy}=\frac{10}{99}\log(10) $$ ...
1
vote
1answer
109 views

Calculating in closed form $\int_0^1 \log(x)\left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$

What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $ would you like to recommend? I'm not against them, they might ...
-2
votes
2answers
39 views

How to write the integral $\int_R 5(x+y)\ dy\ dx$ where $R$ is the region bounded by $y=\frac{1}{7}x$, $x=6$ and the $x$-axis?

I have the integral $$\int_R 5(x+y)\ dy\ dx$$ where the region $R$ is bounded by $y=\frac{1}{7}x$, $x=6$ and the $x$-axis. I don't know how to write this problem exactly. Could anyone edit my ...
0
votes
3answers
21 views

Double integrals volume

Find the volume below $z = 5+3y$ above the region $−5 \leqslant x \leqslant 5$, $0 \leqslant y \leqslant 25−x^2$. How do I solve this? I don't know how to make equation to solve this problem. Anyone ...
4
votes
7answers
131 views

Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $

What is an appropriate method to approximate $$I=\int_0^\infty e^{-t^2 /100} \sin \pi t \ dt?$$ This is for a Physics problem, but in fact I need this in general, as my professor and book taught us ...
11
votes
2answers
170 views

Integral $\int_0^1\frac{\log(x)\log(1+x)}{\sqrt{1-x}}\,dx$

I'm trying to evaluate this definite integral: $$\int_0^1\frac{\log(x) \log(1+x)}{\sqrt{1-x}} dx$$ It's clear that the result can be expressed in terms of derivatives of a hypergeometric function with ...
2
votes
1answer
47 views

Lim sup/inf of average value

Consider $$f(t)= \frac{1}{t} \int_{0}^t \sin(e^s) ds.$$ What is $$\mathrm{lim \ inf}_{t \rightarrow \infty} f(t)$$ and $$\mathrm{lim \ sup}_{t \rightarrow \infty} f(t)?$$ Using $u$-substitution, ...
1
vote
2answers
46 views

How to integrate a function of form f (x)/g (x) $\frac {(ax^n+b) dx}{cx^m+d}$

How to find $\int \frac {(ax^n+b) dx}{cx^m+d}$ (where m>n and m,n are natural numbers,and a,b,c,d are integers) ?
0
votes
2answers
69 views

Calculating integral using Stokes theorem and directly

Here is my task: Calculate directly and using Stokes theorem $\int_C y^2 dx+x \, dy+z \, dz$, if $C$ is intersection line of surfaces $x^2+y^2=x+y$ and $2(x^2+y^2)=z$, orientated in positive ...
2
votes
0answers
39 views

Help Integrating $I=\int\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$

I am trying to integrate the following function involving the Normal CDF ($\Phi$). I actually need the definite integral $$\int^b_a\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$$ for $q+ra,q+rb >0$ but ...
0
votes
4answers
24 views

Double Integral Set Up

The question was stated as follows, Evaluate the following double integral; $$ \iint_R x^3y dA $$ where R is interior of triangle with vertices (0,0), (1,0), & (1,1) . I thought for these ...
0
votes
0answers
34 views

Can I compute marginal distribution this way?

I have posted the same question in the Internet another website. But I did not get the answer replies. I only can come here to have a try. The math statement I put here may not be correct. You can ...
0
votes
0answers
50 views

Contour integration from zero to infinity

When solving an improper integration from $0$ to $\infty$ which involves an even function, the integration limits can be extended from $-\infty$ to $\infty$. For example consider even function $f(x)$; ...
1
vote
0answers
19 views

Residue Theorem on an integral contains a Hankel function and a cosine function

I am trying to solve below integration; $$\int_{0}^{\infty} H_{0}^{1}(pR)\sin(pR)\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. This is related to the question link. Below shows my approach to get ...
1
vote
4answers
137 views

$U_n= \int_{0}^{1}\frac{1}{1+x^{n}}dx$

$U_n= \int_{0}^{1}\frac{1}{1+x^{n}}dx$ where Find $\lim_{n\to \infty} U_n$ can i enter the limit inside ? $W_n= \int_{0}^{1}\frac{x^n}{1+x^{n}}dx$ and i established this relation by parts: $W_n= ...
-1
votes
0answers
20 views

How to perform the following integration using dblquad in MATLAB

I am trying to perform the following integration in MATLAB \begin{equation} \begin{split} F &= @(x,y)(e^{(-0.5([x - \mu_1 \hspace{5pt}y-\mu_2])\Sigma^{-1}([x - \mu_1 ...
0
votes
0answers
30 views

Help with a Lebesgue integration problem

Let $(X,m,\mu)$ be a Lebesgue measure space and $f:(X,m)\to[0,\infty]$ be a Lebesgue measurable function such that $\int_X f d\mu=3$. For each $n\in N$ consider the function ...
1
vote
2answers
20 views

Determine region of integration for homework problem

For my Calc III homework problem, I am unsure how to determine the region of integration. The problem is fairly simple. $$ \text{Find the volume of the solid bounded by the planes }x = 0\text{, }y = ...
2
votes
1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
1
vote
0answers
16 views

Double integral with triangular region of integration

I'm attempting to do a Calc III homework problem, and I feel like I'm on the right track, but somewhere I either mess up or set up the problem incorrectly and I don't know how or why. Here is the ...