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
20 views

How to show this integral (Error function)

I'm given this question. Show that $$\int_{0}^{0.25}\frac{1}{\sqrt{x}}e^{-x}dx=\int_{0}^{0.5}2e^{-u^2}du$$. As I know this integral is an error function. How to show? Can anyone give me some hints? ...
1
vote
1answer
19 views

Volume of solid by Cartesian, Cylindrical, & Spherical

I am having trouble just setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. a) Use integration with Cartesian ...
4
votes
1answer
55 views

Why are the differentiation/integration rules what they are?

So I understand what rules you use where, and the general forms of the rules like: $$\left(\frac{d}{dx}\right)^nx^k=\frac{k!}{(k-n)!}x^{k-n}$$ My question is why are these the formulas that give us ...
5
votes
2answers
89 views

Evaluate $\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$

I want to find a quick way of evaluating $$\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$$ This problem appeared on the qualifying round of MIT's 2014 Integration Bee, which leads me to think ...
1
vote
1answer
23 views

Double integral variable change help

I'm having a tough go with this problem. $\iint \frac{x^2}{y^3} dA$ , Integrate using a change of variables over the region defined inside the curves $y=2x,\; y=x,\; y=x^2,\; y=2x^2$ . I graphed it ...
1
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1answer
29 views

Double Integral Change of variable help

I am having some trouble getting this problem set up, and would appreciate any help. Problem: $\iint \frac{1}{(x+y)^2} dA$. Integrate using change of variables over the region inside the lines ...
0
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1answer
26 views

Volume of solid of rotation about x-axis

Rotation of the region bounded by $x=2y^2-1,\; x=y^2$ and $x$-axis about $x$-axis. I draw out the graph, and found intersection is at $(1,1)$ and $(-1,-1)$ So is it correct to continue doing by ...
0
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0answers
30 views

Hardy Littlewood Circle Method

I'm working through Vaughan's book on the Hardy Littlewood circle method, which uses the following lemma: Suppose that $\alpha \geq \beta$ are positive real numbers, and that $\beta \leq 1$. Then: $ ...
-1
votes
0answers
21 views

Renewal equation

I have a question on renewal equation. $$m(t) = F(t) + \int_0^t (m(t − x)dF(x))$$ can someone tell me how to compute the integral part? it's better if you can tell me step by step until we get the ...
0
votes
2answers
68 views

Evaluate $\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$

My question is: Evaluate $$\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$$ for $k=0,1,...,(n-1)$ and $n \in \mathbb{N}$. I've tried integration by parts but without much success. Any ...
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4answers
62 views

Evaluating $\int_{-a}^{a}\sqrt{a^2-x^2}dx$

Question: How to evaluate $$\int_{-a}^{a} \sqrt{a^2 - x^2} dx$$ This came up while trying to prove that the area of an ellipse is give by $\pi a b$ where $a$ and $b $are the major and ...
0
votes
1answer
36 views

The existence of anti-derivatives

The only thing I can think of is that the function is continuous hence the anti derivative exists. I was wondering if there is anything else that needs to be done/said?
0
votes
1answer
53 views

Primitive of the function $(\sin x)/x$

I know that for some functions, for instance $f(x) = e^{-x^2}$, there does not exist a primitive. Does there is a primitive for the function $f(x) = \frac{\operatorname{sin}(x)}{x}$?
1
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1answer
32 views

Convergence of $\int_0^1 \frac{\ln(1-x)\sqrt{x-x^2}}{\sin(\pi x)} \, dx$

I have difficulties with convergence of this integral: $$\int_0^1 \frac{\ln(1-x) \sqrt{x-x^2}}{\sin(\pi x)} \, dx$$ I found similar problem here Covergence of integral but I don't get the solution ...
2
votes
1answer
22 views

Green's Theorem and limits on y for flux

I'm working through understanding the example provided in the book for the divergence integral. The theorem (Green's): $$ \oint_C = \mathbf{F}\cdot \mathbf{T}ds = ...
1
vote
0answers
17 views

How can we prove $\int_{B_\rho(x_0)}\Delta u\;d\lambda_n=\int_{\partial B_1(0)}\frac{\partial u}{\partial\rho}(x_0+\rho\omega)\rho^{n-1}\;d\omega$?

Let $B_r(x_0)$ and $\overline{B}_r(x_0)$ be the open and closed ball in $\mathbb{R}^n$, respectively $u\in C^2(B_r)$ and $\rho\in (0,r)$ $\lambda_n$ be the Lebesgue-measure on the ...
4
votes
4answers
77 views

How to evaluate $\lim _{n\to \infty }\:\int _{1/(n+1)}^{1/n}\:\frac{\sin\left(x\right)}{x^3}\:dx$?

We have to evaluate the following limit: $$\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx,\:n\in \mathbb{N}$$ First step I wrote that $\int ...
0
votes
0answers
11 views

Continuity of integral from x to x+1 of Lp function

For $1 \le p < \infty$ and $f \in L^p({\bf R})$ define $g(x) = \int_x^{x+1} f(t) dt$. How do I shew that $g$ is continuous? In the case $p = 1$, we have $|g(x) - g(y)| \le \int_{y}^{x} |f(t)| dt ...
0
votes
3answers
53 views

Problem in indefinite integral. (Exponential)

I'm given this integral to integrate. I've no idea where to start with. Perhaps someone can give me some hints or guide me. Thanks a lot. $$\int\frac{(x^3)e^{x^2}{}}{x^2+1}dx$$
1
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0answers
19 views

How would i find the volume of a cone in the $interval [0,a]\times[0,a]\times[0,a]$ and how it's surface area? (using integration?)

whichEssentially i want to find the measure of $z^2\leq x^2+y^2$ and $z^2=x^2+y^2$. Now i know for one of them i would incorporate cilindrical coordinates: $$g(r,\phi,z)=(rcos \phi, r sin\phi,z)$$ ...
0
votes
2answers
18 views

How to find volume of the given solid analytically?

Here is the question - I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help. Edit: The ...
3
votes
1answer
35 views

Two numbers are chosen at random over the interval $ [0,1]$

Two real numbers, $x$ and $y$ are chosen at random over the interval $ [0,1]$. What is the probability that the closest integer to $\frac{x}{y}$ will be even? Floor functions don't place nicely with ...
6
votes
0answers
50 views

Integral Inequality $\leq n^{3/2}\pi$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $ \int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$ It's easy to see ...
0
votes
3answers
40 views

Struggling with integration/differentiation

quick question as I'm sure this is simple but it has me stumped. I have to integrate and differentiate this equation. Not sure on the exponential, had a couple of goes but it doesn't look right. ...
1
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0answers
40 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
-3
votes
2answers
29 views

Proving that a particular integral is a rational number [on hold]

I have the sequence $\left\{ I_n \right \}_{n\geq 0}$ given by: $$I_n=\int_{0}^{1}\frac{\left ( x^2+x+1 \right )^n - 1}{x^2+1}\,dx$$ and I have to prove that $I_{4n+1} \in \mathbb{Q}$ .
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1answer
28 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
0
votes
1answer
12 views

Reasoning behind method of steepest descent

I am considering the method of steepest descent from my notes. I have written that $$\int_a^b dx e^{g(x)} \sim e^{g(x_0)} \int_{\infty}^{\infty}dx \exp \left[-\frac{1}{2}(x-x_0)^2|g^"(x_0)|\right] ...
0
votes
3answers
43 views

Arc-length of an Archimedean Spiral

I want to calculate the arc-length of the archimeadean spiral given by the equation: $\vec{x}(t)=\begin{pmatrix} e^{-\alpha t} \cos t \\ e^{-\alpha t} \sin t\end{pmatrix}$ $\alpha >0$ and $t \in ...
1
vote
1answer
49 views

Find this integral $I(x)=\int_{0}^{+\infty}\frac{1}{y}e^{-y-\frac{x}{y}}dy$

Find this integral $$I(x)=\int_{0}^{+\infty}\dfrac{1}{y}e^{-y-\dfrac{x}{y}}dy$$ I think $$I'(x)=-\int_{0}^{+\infty}\dfrac{e^{-y-\frac{x}{y}}}{y^2}dy$$ Now I have no idea of how to continue
1
vote
1answer
33 views

Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
0
votes
4answers
76 views

Area of the curve sin(cos(x)) [on hold]

Find the area of the region enclosed by the curves $y = \sin (\cos(x))$, $y = 0$ ,$x = π / 2$, and $x = −π / 2$. I am not able to integrate the function. How do I find this area?
5
votes
2answers
41 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
2
votes
1answer
36 views

Riemann Integral on $\mathbb{R}^2$

I have the following question. Find a function $f(x,y)$ that is integrable on rectangle $[0,1] \times [0,1]$, such that $g(y) = f(\frac{1}{2}, y)$ is not integrable for $y \in [0,1]$, or prove that ...
1
vote
1answer
19 views

Integration about x and y axes to find area

I have a problem statement that requires me to find area between the curves about x axis and about y axis. But my answers are not matching. Please find below my worked out solution - The ...
2
votes
1answer
17 views

parallelepiped change of variables

I can't figure out how to start this problem. Use a triple Integral to find the mass of a parallelepiped generated by the vectors $$<6,1,2>,\ <3,3,9>,\ {\rm and}\ <2,7,3>.$$ We are ...
5
votes
5answers
111 views

Proving $\int_{0}^{1}\sqrt{\frac{1-x}{1+x}}dx=\frac{π}{2}-1$

Proving $$\int_0^1 \sqrt{\frac{1-x}{1+x}} \, dx= \frac{π}{2}-1$$ My attempt is: I assumed the $1-x=u$ $du =-dx$ $$\int_0^1 \sqrt{\frac{u}{2+u}}\,(-du)$$ here I stoped and I couldn't how to complete ...
1
vote
1answer
61 views

Definite integral involving 2015

Evaluate $$\displaystyle\int_{2}^{2014} \frac{\log \left( 2015 - x\right )}{\log \left( 2015 - x\right ) + \log \left( x - 1\right )} \mathrm{d}x$$ I got the solution using software, and it is a ...
0
votes
0answers
46 views

how to solve this integral involving any square root

how to solve the integral $\int\sqrt{\alpha+\beta e^{\gamma t}}dt$ i got this integral from the problem Given that the velocity $v$ of a body $t$ segonds after passing a point $O$ is found by ...
5
votes
2answers
72 views

computing an integration with a floor function

I am trying to compute $$\int_0^1 \left(\frac{1}{x} - \biggl\lfloor \frac{1}{x}\biggr\rfloor\right) dx$$ with no success. Any hints?
8
votes
2answers
110 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
0
votes
3answers
82 views

Integrating $f(x) = 1/x$ from $x=a$ to $x=\infty$

Can the integration of $f(x)=1/x$ from $x=a > 0 $ to $x=\infty$ ever be finite? That is, can $\int_{x=a}^{\infty} 1/x$ be finite?
2
votes
5answers
131 views

Is it possible to calculate for example $\int_{0}^{1} x \mathrm{d}2x$

My question is just for fun, but I want also to verify if I understand something in variation calculus... I want to know if it is possible to calculate this : $$ \int_{0}^{1} x \mathrm{d}2x $$ A ...
1
vote
2answers
24 views

Difference between Path, Curve, Graph and Trace

I am having difficulties in understanding the differences between these concepts. We have a new lecturer who loves writing down things in dense mathematical notation (I don't think that's bad but I am ...
4
votes
5answers
79 views

Something wrong at $\int \frac{x^2}{x^2+2x+1}dx$

I have to calculate $$\int \frac{x^2}{x^2+2x+1}dx$$ and I obtain: $$\int \frac{x^2}{x^2+2x+1}dx=\frac{-x^2}{x+1}+2\left(x-\log\left(x+1\right)\right)$$ but I verify on wolfram and this is equal with: ...
2
votes
1answer
58 views

Arithmetic mean of $L^2$ function is $L^2$

I have found the following problem, to which I do not find the solution: Consider $f(x), x > 0$ a function such as $$ \int_0^\infty f^2(x) dx < \infty $$ and let $g(x) = \frac 1x \int_0^x ...
0
votes
0answers
24 views

Confusion about Flow Integral

I am asked to calculate the flow integral $$\int{\vec{F}\cdot\hat{T}ds}$$ of $\vec{F}=<2x,-3y>$ along the fourth quadrant path from (5,-3) to (8,0) along the curve $x^2-10x+y^2+16=0$. So I ...
0
votes
0answers
22 views

Area between curves (Calc) [on hold]

How do you know whether to solve with respect to y or respect to x? I know how to do the rest of the problem, I never know which one to do though.
0
votes
1answer
34 views

How to integrate this using tan(x/2) substitution?

How do I integrate cos(x)/(sqrt(5)+cos(x)) ? I have been advised to use t = tan(x/2) substitution but ended up with a polynomial of degree 4 over one of degree 6 to integrate, which did not have an ...
0
votes
0answers
28 views

taking integral of odd functional form

I'm currently working on a paper and have come across the following integral: $$\int \frac{f'(x)}{(x - f(x))} dx$$ I'm not familiar with any typical way of evaluating the integral and plugging it ...