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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1answer
73 views

Exchanging limits and Riemann Integral

Consider a function $f:\Theta \subseteq \mathbb{R}\rightarrow \mathbb{R}$ continuous at $\theta=\theta_0$ such that $f(\theta)\geq 0$ $\forall \theta \in \Theta$. Consider a sequence of real numbers $\...
4
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1answer
49 views

Calc III Stoke's Theorem Calculate $\int_C v \cdot dr$

The sphere $x^2 + y^2 + z^2 = a^2$ intersects the plane $x + 2y + z = 0$ in a curve $C$. Calculate $\int_C \vec{v} \cdot d\vec{r}$, where $\vec{v} = 2yi -zj +2xk$ So I solved this question by taking ...
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1answer
27 views

Integral of a continuous function of l+1 variables

Consider a function $f(\theta, x): \Theta \times \mathcal{X} \rightarrow \mathbb{R}$ where $\mathcal{X} \subseteq \mathbb{R}$ and $\Theta \subseteq \mathbb{R}^l$. Suppose the map $\theta \rightarrow f(...
7
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0answers
69 views

Prove: $\lim_{h\to0}\int_{-\infty}^{+\infty}|f(x+h)-f(x)|^p\mathrm dx=0$ [duplicate]

Suppose for $\forall [\alpha,\beta]\subset(-\infty,+\infty), f\in\mathcal{R}[\alpha,\beta]$, and $\exists ~p>0$, and $\int_{-\infty}^{+\infty}|f|^p\mathrm dx$ exists.Prove: $$\lim_{h\to0}\int_{-\...
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3answers
73 views

Calculating $\int \sin x e^x \Bbb d x$ by parts twice

Integrate $e^x \sin x $. I know I need to integrate by parts 2 times, but I'm stuck at the second integration. For the first I get $$-e^x \cos x - \int e^x\cdot (-\cos x) \,dx $$ Correct me if I'...
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4answers
107 views

Evaluate the Integral: $\int\frac{dx}{\sqrt{x^2+16}}$

I want to evaluate $\int\frac{dx}{\sqrt{x^2+16}}$. My answer is: $\ln \left| \frac{4+x}{4}+\frac{x}{4} \right|+C$ My work is attached:
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2answers
47 views

Problem involving second degree polynomial

Let $ f : \bf R \to \bf R$ be a polynomial of the form $f(x)=a_0+a_1x+a_2x^2$,$a_2 \neq 0$.If $E_1= \int_ {-1} ^{1} f(x)dx -[f(-1]+f(1)]$,$E_2=\int_ {-1} ^{1} f(x)dx -\frac {1}{2} [f(-1]+2f(0)+f(1)]$ ...
7
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1answer
115 views

Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)

We identify $M_{n}(\mathbb{R})$ with $\mathbb{R}^{n^{2}}$ We put $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu=\lim_{r\to \infty} \int_{D_{r}} e^{-A^{2}}$ where the later is counted as a Riemann ...
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2answers
30 views

Spherical Cap (Integration)

For this question --> I know I need to use Pythagoras theorem but I cannot find where a triangle can be inside of that circle.. I also know the form of the integral is like: $$ Volume = π \int_0^...
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2answers
147 views

Integral $\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$

I need to evaluate this integral: $$\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$$ Apparently, Maple and Mathematica cannot do anything with it, but I saw similar integrals to be evaluated in ...
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2answers
129 views

Will integral be $\frac{\pi}{2}$?

Show that $\int \frac{1-cosx}{x^2}\ dx=\frac{\pi}{2}$. I used Taylor's series for cosx to find integral but I don't see intergal becoming equal to $\frac{\pi}{2}$ without any limits of integration. ...
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0answers
89 views

How to calculate the indefinite integral of $\frac{\sin(x)}{\sqrt x}$?

So my question simply is: what is $\int \frac{\sin(x)}{\sqrt x}\,d x $?
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1answer
74 views

Simpler proof of an integral representation of Bessel function of the first kind $J_n(x)$

While doing research in electrical engineering, I derived the following integral representation of the Bessel function of the first kind: $$J_n(x)=\frac{e^{in\pi/2}}{2\pi}\int_0^{2\pi}e^{i(n\tau-x\...
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2answers
72 views

Are these two naive upper bounds ok?

EDIT: Is my work ok? Here, I am trying to show a uniform bound for the sum of $cos(n)$ $$|\sum_{n=1}^{N} cos(n)|$$ $$=\big |\sum \frac{e^{in} + e^{-in}}{2}\big|$$ $$\le \sum |\frac{e^{in} + e^{-in}...
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1answer
33 views

Family of functions with specific properties

I am wondering if there is possibly a well-known family of functions $f_l:\mathcal{R}\to\mathcal{R}$ parametrized by a single positive real / integer value $l$ that has the following properties: 1). $...
2
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4answers
109 views

Computing the Integral $\int \frac{\sqrt{x}}{x^2+x} dx$

Find $\displaystyle \int \dfrac{\sqrt{x}}{x^2+x} dx$. What would be the best way to integrate this? I saw the answer to this and it looked simple so that might mean the steps would be too?
3
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2answers
47 views

Deriving the limit of an integral

Consider a sequence of functions $f_n: \mathbb{R}\rightarrow \mathbb{R}$ and another function $g:\mathbb{R}\rightarrow \mathbb{R}$, $g\geq 0$. Suppose that $$ \lim_{n \rightarrow \infty}\int_{a}^b (...
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1answer
14 views

Limits of a double integral,

From the above image, I understood how it's solved. However, what's confusing me is the limit of $d = 2-x$, why isn't it $2$?
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1answer
59 views

How can I calculate $\int \frac1{x \cos x}\,dx$ ? / An issue wit an ODE.

How can I calculate this integral? $$\int \frac1{x \cos x}\,dx$$ Actually I was lost in this differential equation $$y' = -\frac{(x+2) \sin y}{x \cos x}$$ so I'd be glad if you could help me ...
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1answer
35 views

Volume of a Sphere Segment Cut Out by Zigzag Motion of Radius Vector Through Sphere Center

A closed loop curve on a sphere radius $R$ enclosing curved area $A$ subtends a solid angle $ A/R^2$ steridians at center of the sphere. Volume $V$ is enclosed by concurrent straight generators ...
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0answers
24 views

Showing limit result using change of variables

Consider the real valued random variables $X$ whose image is the set $\mathcal{X} \subseteq \mathbb{R}$. Suppose $X$ has probability distribution $P_{\theta_0}$ where $\theta_0 \in \Theta \subseteq \...
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2answers
31 views

Definite or indefinite integral

[Beginning calculus question.] From Edwards and Penney (6e) p. 810, worked Example 8: Suppose that a moving point has given initial position vector $\boldsymbol{r}(0) = 2\boldsymbol{i}$, initial ...
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1answer
46 views

Let $S:=\left\{(x,y):|x|+\left|y\right|\le1\right\}$; then how to evaluate $\displaystyle{\int\int_S e^{x+y}\,\operatorname{d}x\,\operatorname{d}y}$?

Let $S:=\left\{\left(x,y\right):\left|x\right|+\left|y\right|\leq 1\right\}$. How to evaluate $$ \displaystyle{\iint_S e^{x+y} \, \operatorname{d}x \,\operatorname{d}y}? $$ Please help. Thanks in ...
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1answer
42 views

interchange of $L^1$ and $L^{\infty}$ norm

Let $x,y \in \mathbb R^d,$ and $0\neq t \in \mathbb R.$ Define $f(y)= \sup_{x\in \mathbb R^d}\{e^{-\pi |y-tx|^2/ (1+t^2)} \}.$ My Question is: Is it true that $f\in L^{1}(\mathbb R^d)$? Is it ...
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0answers
50 views

What are some motivations that led to the development of the Lebesgue integral?

There are many kinds of integrals, with the most famous being the Riemann integral which is taught in elementary calculus classes. The motivation behind the Riemann integral is to find the area ...
3
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1answer
51 views

Find dimensions of a rectangular box that is inscribed in a cone

A rectangular box that rests on the $x, y$ plane is to be inscribed in the cone $z = a − \sqrt{x^2 + y^2}$, $\ a \ge z \ge 0$. Find the dimensions of the box that maximise its volume. How would I ...
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1answer
27 views

Find limits of integral for plane polar co-ordinates question

Use plane polar co-ordinates or otherwise to evaluate the integral $$\int\int_D^\ \frac{x^2-y^2}{x^2+y^2} dA$$ where D is the part of the x,y plane bounded by the parabola $y^2=4(1-x)$ and the ...
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1answer
30 views

Define the function $H:\mathbb{R}\rightarrow\mathbb{R}$ by $H(x)=\int_{-x}^{x}[f(t)+f(-t)]dt\forall x$ Find $H''(x)$.

Suppose that the function $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable. Define the function $H:\mathbb{R}\rightarrow\mathbb{R}$ by $$H(x)=\int_{-x}^{x}[f(t)+f(-t)]dt\qquad\forall x$$ Find $H''...
6
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1answer
71 views

“Increasingify” a function

Let $f : [a,b] \rightarrow \mathbb{R}$ be a $C^1$ function such that $f$ is monotonic on each $[t_k, t_{k+1}]$, with $a = t_0 < t_1 < ... < t_N = b$. Let g be the increasing-ified version of ...
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5answers
100 views

Integrate $4x/(x^4-1)$ dx

I'm having trouble solving the integral $$\int_{5/4}^{13/12}\frac{4x}{x^4-1}\,dx$$ I have a feeling it is to do with the log integration identity but can't seem to manipulate it without involving ...
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1answer
31 views

How would I reverse the order of these integrals?

In the question, x is the first integral at the moment. Its $\sin^{-1}(y^2)<x<\pi/2$. Y is the second integral which needs to become the first. Its $0<y<1$. I know how to integrate the ...
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0answers
39 views

Different forms of remainder in Taylor series

In the literature, it is common to find different forms of the remainder function in a truncated Taylor series. To name a few: Integral form Lagrange form Cauchy form First, can you tell me any ...
4
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1answer
63 views

Calculate $\int_0^1 f(x) dx$, where $f(x)= \begin{cases} 0 & x= x\in C \\ \frac{2}{7^n} & x= x\not \in C \end{cases} $, with $C$ the Cantor set.

I am looking back at notes, and problems from the semester, and I came across this problem that I am having trouble solving. Let $$f(x)= \begin{cases} 0 & x= x\in C \\ \dfrac{2}{7^n} ...
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0answers
30 views

antiderivative of $\psi'(u)$ for $u\in W^{1,2}_0((a,b))$

Let $u\in W^{1,2}_0((a,b))$ and $\psi'$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. If I want to consider the antiderivative of $\psi'(u)$, what happens with the $u$ inside $\Psi(u)$...
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2answers
60 views

If $\ln x$ is integrable, then is $x \ln x$ also integrable?

I have a very simple problem. Assume we have a finite measure $\mu$ on $[1,\infty)$, and \begin{align} \int_1^\infty t ~d\mu(t) < \infty. \end{align} My question is if this implies \begin{align} \...
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1answer
80 views

Compute $\lim_{n\to\infty}\int_0^1\frac{x\sin{nx}}{1+x^2n^6}dx$

$\lim_{n\to\infty}\int_0^1\frac{x\sin{nx}}{1+x^2n^6}dx$ My Work: By the comparison principle: $$\lim_{n\to\infty}\int_0^1\frac{x\sin{nx}}{1+x^2n^6}dx\le\lim_{n\to\infty}\int_0^1\frac x{1+x^2n^6}dx$$...
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0answers
79 views

Fundamental Theorem of Calculus application question

I have a question that I believe I have the solution to, but I am not sure and there is no way of checking. As such I was hoping someone could look over my solutions and provide me with some feedback. ...
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0answers
24 views

Does this integral of Appell F_1 converge?

I'm interested in whether or not integrals of the form $$\int_{0}^{1}\mu^{\alpha}F_{1}\left(\frac{\alpha}{2};1,-1;\frac{\alpha+2}{2};\mu^{2},-\beta\mu^{2}\right)\mathrm{d}\mu$$ converge, and if so ...
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3answers
469 views

The value of double integral $\int _0^1\int _0^{\frac{1}{x}}\frac{x}{1+y^2}\:dx\,dy$?

Given double integral is : $$\int _0^1\int _0^{\frac{1}{x}}\frac{x}{1+y^2}\:dx\,dy$$ My attempt : We can't solve since variable $x$ can't remove by limits, but if we change order of integration, ...
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2answers
76 views

Solving this rather tedious integral

I need help solving this integral $$\left \langle x \right \rangle = \frac{2}{a}\int_{0}^{a}x\sin^{2}\left ( \frac{m \pi x}{a} \right )\,dx$$. I have tried to reduce $$\sin^{2}\left ( \frac{m \pi x}{...
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2answers
80 views

Lipshitz Integral for $a=0$

I knew that this, $$\displaystyle{\int_0^\infty e^{-ax}J_0(bx)dx=\frac{1}{\sqrt{a^2+b^2}}},$$ holds for $a>0$ but, in an exercise from Arfken, it said that this holds for $a\geq0$. How can I prove ...
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3answers
38 views

Using Divergence Theorem to evaluate the integral

Find the value of $$ \iint_{\Sigma} <x, y^3, -z>. d\vec{S} $$ where $ \Sigma $ is the sphere $ x^2 + y^2 + z^2 = 1 $ oriented outward by using the divergence theorem. So I calculate $ div\vec{F}...
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1answer
74 views

Determine poles and residues of contour integral using Laurent series

I want to find the residues of the integral $F = \int_{-\infty}^{\infty} \dfrac{1}{x+(a-ib)} \dfrac{1}{\exp(-x/c)-1} dx$ I know that $x=-(a-ib)$ is a simple pole which contributes a non-zero residue....
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2answers
38 views

Taking the derivative of an integral using chain rule

$2 \frac d {dy} (\int_0^{\sqrt y}3x^2 dx) $ I know that this gives you $3y^{\frac 1 2}$ as a result, if done step by step, but I've been told I can use chain rule to to do it in a single step. I've ...
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1answer
61 views

How can one derive the circumference of a circle using integrals?

Many proofs for the area of a circle start with something like $$ A(r) = \int_0^r 2 \pi t dt $$ such as at https://en.wikipedia.org/wiki/Area_of_a_disk#Onion_proof , but I don't understand how to ...
4
votes
2answers
43 views

Integrate $\cos(x^3)$ over given bounds

I'm not really sure at all how to integrate this function. I was wondering if anyone could help me out, $$\int^4_0 \int^2_{\sqrt y} \cos(x^3) \, dx \, dy$$ The answer choices are ${1 \over 3}\sin(...
1
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1answer
29 views

Convergence of Definite Improper Integrals of the Form $1/x$

Given a simple integral of the form: $$ \int ^1 _{-1} \frac{1}{x} \, dx =\lim _{a\rightarrow 0} \int ^1 _a \frac{1}{x} \, dx + \int ^a _{-1} \frac{1}{x} \, dx$$ Is it possible to say that this ...
0
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1answer
41 views

Initial value problem without explicit constant finding

So I was reading my professor's differential equation book until I came across a weird way of calculating a initial value problem without explicity having to calculate the constant that made that ...
2
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1answer
41 views

Use The Divergence Theorem to evaluate the flux

Let $ \vec{F}(x, y, z) = (\sin z + xy^2)\vec{i} + x^2e^{3z}\vec{j} + (\cos ^3x + x^2z)\vec{k}. $ Let $ T $ be the surface bounding the region $ R $ given by $ x^2 + y^2 \leq z \leq 6 - \sqrt{x^2 + y^...
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2answers
48 views

Newtons Law of Cooling (Integration)

The question is : An apple crumble is taken out of the oven at 7:30pm. At that time, its temperature is 100 degrees Celsius. At 7:40pm, ten minutes later, the temperature of the apple is 80 ...