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
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1answer
36 views

Differentiation under an integral with respect to a function

Consider the functional $F$ defined via the integral $$ F(\mu)=\int_0^\ell\int_0^\ell f(s,t)\mu(s)\mu(t)\,ds\,dt. $$ How would I differentiate this with respect to $\mu$? I realize that this has ...
3
votes
1answer
42 views

Riemann integral and Lebesgue integral (in real analysis Folland)

The following is from p.57(Real Analysis, by Folland) My question is following: Dominated convergence theorem should roughly be: $f_n \rightarrow f, |f_n|\leq g$ a.e., then $\int f = ...
2
votes
0answers
51 views

Find average value of function over tetrahedron

The question is: Find the average value of $f(x, y, z) = x + y + z$ over the tetrahedron with vertices $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, and $(1, 1, 1)$. I know how to find average value but am ...
1
vote
2answers
38 views

Assistance on beginning the integral $\int\frac{dx}{(x+1)(n-x)}$ [duplicate]

This is the integral $$\int\frac{dx}{(x+1)(n-x)}=\int kdt$$ I just need some assistance on how to begin the left side integral and I will most likely be able to continue it from there thank you.
2
votes
1answer
265 views

Prove that :$\frac{1}{100\pi}>\int_{100 \pi}^{200\pi}\frac{\cos(x)}{x}>0$

enter image description here Using integration by parts prove that $\frac{1}{100\pi}>\int_{100 \pi}^{200\pi}\frac{\cos(x)}{x}>0$. Could anyone give me a help with this problem? I have tried ...
1
vote
1answer
34 views

Integral syntax

I've come across the following formula. $ [rx_csinθ - ry_ccosθ + r^2θ]^{θ2}_{θ1} $ Does this equate to the following? $ (rx_csinθ_2 - ry_ccosθ_2 + r^2θ_2) - (rx_csinθ1 - ry_ccosθ_1 + r^2.θ_1) $ I ...
3
votes
2answers
37 views

How would I take this derivative?

I am not really sure how I would take the following derivative: $\frac{\partial}{\partial r}\left( F(r) \right) = \frac{\partial}{\partial r}\left( \int_{0}^{2 \pi} f(r,\theta) d\theta \right)$ ...
2
votes
2answers
87 views

How to calculate $\int xe^{1/x^2} \ dx$

How would I go about integrating $\int xe^{1/x^2} \ dx$? I attempted a u-substitution and integration by parts already, but that didn't get me anywhere.
2
votes
2answers
56 views

Calculating $\iint (x+y) \, dx \, dy$

By using the change of variable $u=x+y$ , $v=x$ evaluate $$\iint_{Ta} (x+y) \, dx \, dy$$ where $Ta$ is the region in the $xy$ plane bounded by the $x$ and $y$ axes and the line $x+y = a$. Update: I ...
1
vote
1answer
32 views

If $0<\mu(X)<\infty$ and $0<p<q< \infty$. Is $(\int_X |f|^p) ^\frac{1}{p}\leq \int_X |f|^q d\mu) ^\frac{1}{q}$ true?

If $0<\mu(X)<\infty$ and $0<p<q< \infty$ Is $(\int_X |f|^p d\mu) ^\frac{1}{p}\leq \int_X |f|^q d\mu) ^\frac{1}{q}$ true. I should say that $\frac{1}{p}+\frac{1}{q}=1$ I think it is ...
3
votes
0answers
43 views

Integration by change of variable $\iint (x+y) \, dx \, dy$ [duplicate]

I am not really sure how to tackle this one. Any help is appreciated! By using the change of variable $u=x+y$ , $v=x$ evaluate $$\iint_{Ta} (x+y) \, dx \, dy$$ where $Ta$ is the region in the $xy$ ...
0
votes
0answers
73 views

Integration by part

Trying to simplify this integral by using integration by parts. $$ \int\limits_0^1 \int\limits_0^{g(x)} y.f(x,y) dydx $$ I define $$ u=y \space \space and \space \space dv=f(x,y)dy$$ $$ du=dy ...
0
votes
2answers
35 views

finding the integral with substitution [closed]

I have to find the integral of a function and I am not sure about the beginning of the integral. I have solved it but I would like to know why the following procedure happens $\int x^7 ...
3
votes
1answer
36 views

Definition of integration of differential forms

I am trying to understand precisely the following paragraph: Question Why would he define the support $K$ of a form $\omega$ defined on an open set $U$ as a subset $K\subseteq M$ instead of a ...
0
votes
1answer
29 views

The first and second derivative of a triple integral

I need the first and second derivative of a triple integral function, and f(x,y) is a density function and differentiable $$ w(v) = \int\limits_0^v \int\limits_0^s \int\limits_0^{u(x)} f(x,y) dy dx ds ...
2
votes
2answers
40 views

Does there exist function that behaves in this way around some point and is continuous at that point?

Suppose that we have some function $f: \mathbb R \to \mathbb R$ such that $f$ is integrable (Riemann or Lebesgue, choose one, or some other maybe more general type of integration, if there is such) on ...
2
votes
1answer
41 views

Calculate $ \int_0^\infty \! exp(-\frac{1}{2}x^2)x^k \, \mathrm{d}x $

$k \in \mathbb N$. I know, that this is an odd function, if $k$ is an even number. So I have to calculate the integral from $-\infty$ to $+\infty$ and that would be $0$ for $k$ as an even number. But ...
2
votes
2answers
63 views

Why this doesn't contradict Monotone and Dominated Convergence Theorem? [closed]

$$\lim_{n \to \infty} \int_0^\infty f_n(x)dx \ne \int_0^\infty \lim_{n \to \infty} f_n(x)dx$$ where : $f_n=ne^{-nx}$ for all $x \in [0,\infty)$ $n \in \mathbb{N}$ Can somebody help me with ...
0
votes
0answers
20 views

Solving a weighed integral equation

I am struggling in solving a specific type of equation which simply pops out when we're dealing with weighed functions. I think the general context is worth mentioning. Let's say we have a discrete ...
2
votes
2answers
65 views

Assymptotic approximation of $\int x^n \log xdx$

Hi I'm following an Analysis course and one of the exercises is to prove that: $$\int_{1/2}^1 x^n \log x dx \approx -1/n^2 $$ $$\left|\int_0^{1/2} x^n \log x dx\right| = O\left((1/2)^n\right)$$ At ...
1
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0answers
53 views

Integrating path of a diffusion process

Let $X_t$ be a process that satisfies the Ito diffusion process: $$dX_t = a(X_t) dt + b(X_t) dW_t$$ I am interested in approximate numerically $$I(\omega) = \int_0^t X_s ds$$ and eventually also ...
0
votes
1answer
51 views

$F(x) = \int_x^{x+1} \sqrt {\arctan {t}}dt$ is bounded for $x \ge 0$.

$F(x) = \int_x^{x+1} \sqrt {\arctan {t}}\space dt$ is bounded for $x \ge 0$. My attempt is $\sqrt {\arctan {t}}$ is continuous, therefore Riemann integrable and so $F(x)$ is continuous. $F(x)$ is ...
0
votes
2answers
74 views

Finding Upper and Lower limits using four vertices of a Trapezoid

I need a little help with finding upper and lower limits of the below equation using four vertices of a Trapezoid. The four vertices are upper left$(1,3)$, upper right$(5,3)$, lower left$(2,1)$, and ...
1
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0answers
20 views

Is $\int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 \cdots \int_{[(n+1)/2-\sum_{k=1}^{n-1} 1/x_k]^{-1}}^2 \prod_{k=1}^n f(x_{k-1},x_k) dx_1 \dots dx_n \geq cq^n$?

Let $f\colon (1,2)\times (1,2) \to \mathbb{R}$ be a Lebesgue measurable, bounded and non-negative function such that $$ \int_1^2 f(x,y) dy = 1, \qquad x \in (1,2). $$ Moreover, assume that for any ...
2
votes
2answers
43 views

Question 45 in Chapter 19 in Michael Spivak's book “Calculus” involving an improper integral

This is Problem 45 in Chapter 19 in Michael Spivak's book "Calculus". (a) Suppose that $\frac {f(x)} x$ is integrable on every interval [a, b] for $0$ < a < b, and that $\lim_{x\to0}f(x)=A$ ...
2
votes
2answers
54 views

$\int_0^{2\pi}\frac{1}{a\cos \theta+b\sin\theta+d}d\theta$ where $a,b,d\in\mathbb{R}$ and $a^2+b^2<d^2$

$\int_0^{2\pi}\frac{1}{a\cos \theta+b\sin\theta+d}d\theta$ where $a,b,d\in\mathbb{R}$ and $a^2+b^2<d^2$ Here, I solve it by Residue Theory. By substituting $d\theta=dz/iz,\cos ...
4
votes
2answers
119 views

Evaluating $\displaystyle\int\frac{x^4}{x-1} \, dx$

This question involves long division. I calculated the value. However, I want to ask two concept questions: 1) Why am I doing long division rather than writing out the form of the partial fraction ...
0
votes
1answer
46 views

How to solve the Integral $\int_{-\infty}^\infty [\tanh(\frac{x+a/2}{b})-\tanh(\frac{x-a/2}{b})]e^{ikx} dx$?

How to do the integral $$\int_{-\infty}^\infty [\tanh(\frac{x+a/2}{b})-\tanh(\frac{x-a/2}{b})]e^{ikx} dx$$ I had deal with this problem for whole day long...... I tried to turn the $\tanh$ into ...
1
vote
1answer
68 views

Complex Integral Over an Ellipse

Suppose we had a function defined as $$ S(\zeta) = \zeta - \sqrt{\zeta^2 - c^2} $$ and we wish to evaluate $$ \oint_\gamma \frac{S(\zeta)}{z-\zeta}\, d\zeta, $$ where $\gamma$ is the (positively ...
3
votes
1answer
48 views

Derivative of the integral with respect to the function

Consider this function: $$ E[L] = \int\int\{ y(x) - t \}^2p(x,t)dx dt $$ I try to figure out how to take the derivate of this function with respect to $y(x)$. In the book it is: $$ \frac{ \delta E[L] ...
1
vote
1answer
65 views

Difference between ordinate and abscissa.

While explaining the use of integrals in finding the area under a curve y=f(x), my book gives the following explanation : We can think of the total area A of the region between x-axis, ordinates x ...
2
votes
1answer
38 views

A guess about Jacobi

Let $\Omega \subset R^n$ is a open set of $R^n$ , $f:\Omega\rightarrow R^n $ is $C^2$ function.The Jacobi of $f$ is : $$ J_f(x)=\frac{D(f_1,...,f_n)}{D(x_1,...,x_n)}=|\frac{\partial f_i}{\partial ...
3
votes
2answers
102 views

A Bound for the Error of the Numerical Approximation of a the Integral of a Continuous Function

How to numerically integrate a nasty function? Suppose $f$ is only continuos; which method can you employ to approximate $$\int_0^t f(s)ds$$ Since $f$ is continuos the integral exists, but all ...
0
votes
1answer
45 views

Integral that depends on itself

I have a vertical string hanging under its own weight so $v(t)=\sqrt{T/ \rho}=\sqrt{(l-s(t))g}$. If I want to find the distance traveled by a pulse in a time $t$ is get $s(t)=\int_0^t \sqrt{(l-s(t))g} ...
1
vote
1answer
22 views

When doing an area between curves, what is a quick way to know which is the top and bottom & left and right graph?

I'm taking calculus and we're up to areas between curves. Thing is that unless I do a table of values and graph, or I'm given an easy transformation, its really hard to figure out which graph is the ...
0
votes
1answer
44 views

Fatou's Lemma in nonpositive function

Fatou's Lemma: If $\{f_n\}$ is a sequence of non-negative measurable function, then $\int(\text{lim inf} \ \ f_n)\leq \text{lim inf}\int f_n$. If $\{f_n\}$ is a sequence of non-positive ...
3
votes
2answers
56 views

Integration ambiguity

I am trying to Evaluate $$I=\int\frac{x^4+1}{x^6+1}dx$$ Now $I$ can be written as $$I=\int\frac{x^4(1+\frac{1}{x^4})}{x^6(1+\frac{1}{x^6})}dx=\int\frac{(1+\frac{1}{x^4})}{x^2(1+\frac{1}{x^6})}dx$$ ...
1
vote
1answer
39 views

Why direction of the curve does not matter in linear integral in respect to arc?

So i know that the linear integral in respect to arc is the area and no matter which direction we go with the curve we will get the same result. But could anyone show me a proof that does not rely on ...
1
vote
1answer
26 views

Multiple Integration - Evaluating Volume of Solid

We are asked: Evaluate the double integral by first identifying it as the volume of a solid. The problem is as follows: $$\iint_R (5-x) \, \mathrm{d}A$$ where $$R=[0,5]\times[0,3].$$ The answer ...
0
votes
1answer
20 views

Multiple Integrals - Evaulating as Volume of Solid

We are asked: Evaluate the double integral by first identifying it as the volume of a solid. The problem is as follows: $$\iint_R 3 \, \mathrm{d}A$$ where $$R=[-2,2]\times[1,6].$$ The answer is: ...
3
votes
1answer
64 views

Definite integral, Green's function

how would you tackle this integral: $$\int_0^{2\pi} \frac{\sin^3 x}{1+r^2-2r\cos(x-5)}dx$$I presume some of you might notice the integrand (at least the denominator) is the Green's function - nearly, ...
1
vote
3answers
60 views

What is a solution to this integral?

What is a solution to the following integral: $$\int_0^\infty \, \frac{\cos{kt}}{\pi}\,\mathrm{d}k\,?$$ I have tried to evaluate this in the usual way: ...
0
votes
1answer
25 views

Integral that results in the fraction of two gamma functions

I'm trying to show this equation $$ \int\limits_0^\infty \mathrm{d}x_1 \dots \mathrm{d}x_n \left( 1 - \sum_{i=1}^n x_i \right)^k \Theta \left( 1 - \sum_{i=1}^n x_i \right) = ...
0
votes
1answer
40 views

Question about Integration of Multiplication of Two functions

If we have the following conditions and $g(x)\geq 0,\; \forall x \in [0, x_0]$ and $f(x) < 0,\; \forall x \in [0, x_0]$ and $\int_{0}^{x_0}g(x)dx = 1$. How can I show that if it is? ...
3
votes
0answers
79 views

Integral of $e^{ix^2}$

How does one evaluate $$\int_{-\infty}^{\infty} e^{ix^2} dx$$ I know the trick how to evaluate $\int_{-\infty}^{\infty} e^{-x^2}dx$ but trying to apply it here I get a limit which does not converge: ...
2
votes
1answer
49 views

Work done to pump water out of a conical tank into a window above it

Water is pumped from a conical tank of top radius 3 ft and a height of 5 ft to a window 10 ft above the tank. The tank is completely full of water. How much work is done? This is what I have so ...
1
vote
0answers
44 views

evaluation or simplification of integrals

i need a help to evaluate or simplify as possible the following integrals , my aim is find an approximation to this integration : here $$m(x)=\int_0^x \frac 1 {\sqrt{2\pi t^3}}\exp\left(-\frac 1 ...
1
vote
1answer
35 views

Convergence of stochastic processes and integral

Let $X_n(t)$ and $X(t)$ are bounded stochastic processes ($|X_n(t)|<C$ and $|X(t)<C|$) such that for each $t\in[0,T]$ $$ X_n(t) \to X(t) \text{ as } n\to\infty \text{ almost surely }. $$ ...
0
votes
3answers
43 views

Evaluating $f(x) = f(0) + xf'(0) + \int_{0}^{x}f''(t)(x-t)dt$

Suppose that a function $f$ has derivatives of all orders near $x=0$. By the Fundamental Theorem of Calculus: $f(x)-f(0)= \int_{0}^{x} f'(t)dt$ a. Evaluate the integral using ...
1
vote
2answers
68 views

Understanding this integral from a measure theory perspective.

I know that $$\int_1^\infty \frac{1}{x} dx$$ does not converge in regular calculus. But I'm looking at $L^p$ spaces now and this integral is a good counter example for some things, but what is the ...