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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Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $\int_a^b {\frac{dx}{\sqrt{|p(x)|}}}\int_b^c {\frac{dx}{\sqrt{|p(x)|}}}$

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $$\int_a^b {\frac{dx}{\sqrt{|p(x)|}}}\int_b^c {\frac{dx}{\sqrt{|p(x)|}}}$$ My attempt: I perform linear substitution $u=x-a+c$ ...
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1answer
27 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
2
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3answers
51 views

Real Methods to Evaluate $2 \int_{-1}^{1}x^2 \sqrt{1-x^2}dx$

I was recently contacted by a friend to find the values of the two following integrals by any means. $$ I=2\int_{-1}^{1}x^2 \sqrt{1-x^2}dx$$ $$ J=\int_{-1}^{1}(1-x^2) \sqrt{1-x^2}dx$$ The first ...
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Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
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36 views

What is the integral of $ \int \frac{-6000} { (3x+50)^2} dx$ [on hold]

How can I find the value of the integral $$ \int \frac{-6000} { (3x+50)^2} dx$$
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2answers
31 views

What's my mistake in this integral transformation?

I've the following integral, which should result in 1, as shown by the scetch, but in my calculation I get the result 0. What's my mistake? Sorry the comments are in German and please note that a ...
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12 views

Drawing slices and projections of an iterated integral.

I'm having a rough time visualizing and graphing the slices and projections of this iterated integral: $\int \limits _0 ^1 \int \limits _y ^1 \int \limits _y ^x x \mathbb e ^{z^2} \space \mathbb d x ...
2
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1answer
32 views

Understanding the Definition of a derivative as slope of a tangent line

I'm trying to understand the derivative and am wondering why the derivative is described as the slope of the tangent line and not the slope of a function itself. Say $f(x) = 2x+5$ where ...
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1answer
44 views

Integration by parts tricks

Are there any useful tricks to integration by parts apart from: $$\int f(x) = \int 1 \cdot f(x)$$ $$\int f(x) = ... = g(x) + c \cdot \int f(x) \rightarrow \int f(x) = \frac{g(x)}{1-c}$$ How would ...
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3answers
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Solving with integration by parts: $\int \frac 1 {x\ln^2x}dx$

Solving: $$\int \frac 1 {x\ln^2x}dx$$ with parts. $$\int \frac 1 {x\ln^2x}dx= \int \frac {(\ln x)'} {\ln^2x}dx \overset{parts} = \frac {1} {\ln x}-\int \frac {(\ln x)} {(\ln^2x)'}dx$$ $$\int ...
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1answer
30 views

Understanding Dirac Delta

I found this: here: http://www.nada.kth.se/~annak/diracdelta.pdf on page 2 Can anyone explain how and why all the terms are cancelled in the second step?
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1answer
14 views

$f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable

If $f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable, then $f$ is Riemann integrable and that $\int _c^b f \to ...
2
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1answer
26 views

How do I complete this proof that the absolute value of an integral function is an integrable function?

I'm trying to complete the proof in this answer that if $f: [a, b] \to \mathbb{R}$ is a Riemann integrable function, then $|f|$ is an integrable function also. I understand the proof that $$ ...
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1answer
16 views

How we can find $A_{(\Gamma_f)}$?

We have $f,g:[-4,4]\rightarrow\mathbb{R}$, $f(x)=x^2+2$ and $g(x)=x+4$. We need to find the crowd area between the graphs f and g. I know that $A_{(\Gamma_f)}=\int_a^b|f(x)|dx$ but in this case how ...
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1answer
17 views

Is the product of a Schwartz function and a locally integrable function integrable?

Let $f\in S(\mathbb{R}^n)$ the space of rapidly decreasing functions on $\mathbb{R}^n$ and $g\in L_{loc}^1(\mathbb{R}^n)$. Is $fg$ integrable? Namely is it true that $$ \int |fg| <\infty. $$
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2answers
20 views

$f$ is Riemann integrable

We know that if $f \in \mathcal R[a,b]$ and if $a = c_0 < c_1<\cdots<c_m =b$, then the restrictions of $f$ to each subinterval $[c_{i-1},c_i]$ are Riemann integrable. Is the converse true, ...
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0answers
52 views

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ using elementary, high school techniques [duplicate]

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ $$$$ I was given this integral by a friend who saw this here on MSE. He asked me if I could solve it using the very ...
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Laplace transform of inverse error function

I want to calculate the convolution of a function with the inverse error function. Therefore I chose to try to first find an integral transform of the inverse error function like the laplace ...
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2answers
57 views

Antiderivative of $xe^{-cx^2}$

I need to define $c$ in $$\int_0^\infty xe^{-cx^2},$$ so that it becomes a probability-mass function (so that it equals 1). Where do I even begin finding the antiderivative of this? I know the answer ...
2
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2answers
33 views

nonnegative Riemann-integrable function, infimum

$f$ is a nonnegative Riemann-integrable function on $(0,1)$ and $f(x)\ge\sqrt{\int_0^xf(t)dt}$ for $x\in(0,1)$. Find $\inf\frac{f(x)}{x}$ I have no idea how to work out the assumption, for equality ...
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2answers
59 views

Area between a semicircle and a 45° line

I'm studying for a Calculus test and I met the following question: There's a semicircle $$y=\sqrt{1-x^2}$$ and a line at 45° degrees v=x. The task was to find the area in the positive quadrant. I ...
2
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0answers
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When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
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0answers
19 views

follow-up question to Hake's theorem in Bartle's book

My question is based in here. Why is it that $b$ forces to be a tag of $[x_{m-1},b]$? I can't get the right trick. Can you please give me some hints? Thanks
2
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1answer
66 views

How to calculate the integral of $f(x)$? [on hold]

Let $f(x)$ be a function which satisfies the following two properties: 1) $f(x) + f(-x) =2$ 2) $f(1-x) = f(1+x)$ I need to calculate the $\int_0^{2016} f(x)dx$. I already tried to find $f(x)$ ...
2
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3answers
81 views

Calculate an integral depending on n

Is there a way (simple or not) to calculate the following integral? $$\int_{-1}^{1} \sqrt[n]{1-x^n} dx$$ Thanks
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27 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}^{+}\land0<a<1$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
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1answer
28 views

Integral with an unknown function

I am trying to solve this integral $$ \int \frac{f(x)}{g(x)}\frac{\mathrm dg}{\mathrm dx}\mathrm dx $$ where $g$ is an unknown function of $x$, and $f(x)$ is a known function that can be integrated ...
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1answer
27 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
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0answers
33 views

Weird question about interval of convergence

The question is: if $$ f(x) = \sum \limits_{n=0}^\infty x^n$$ determine the interval of convergence for the power series representation of $$\int_0^x f(t) \, dt$$ That integral threw me off.
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1answer
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Spivak Calculus Ch. 19 #15

(a) Find $\int \sin^4 x\, dx$ in two different ways: first using the reduction formula and then using the formula for $\sin^2x$. (b) Combine your answers to obtain an impressive ...
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How to find volume using Riemann sum with expression $e^x + 3x^3 - x^2$? [on hold]

I'm desperate. It is for math class so please help if you know how to find volume using Riemann sum, and not double integral.
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1answer
26 views

Parametric integral and equivalence in $\infty$

I have to find a equivalent when $x$ comes to $\infty$ for all $a$ (fixed) in $\mathbb{R}_+^*$ of this integral : $$ \int_0^a \frac{e^{-xt}}{\sqrt{a-t}}\mathrm{d}t $$ My work : For $x \in ...
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0answers
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$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$ convergence

Does $$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$$ converge? Domain of this integrand is $x \in \mathbb{R} : x\neq 0, ...
3
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3answers
226 views

double integral $\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
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1answer
17 views

Sequence from generating function with integral

So, let $A(x)$ be the generating function of $a_0,a_1,\dots$ then what would be the sequence of the generating function: $$\int^x_0 A(t)dt$$ Since I am not much acquainted with integrals any help ...
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Infinite sums over integral of triple associated Legendre polynomials

I have a couple of integrals of triple infinite sums of associated Legendre polynomials, which I'd like to integrate using Gaunt's Formula. Any help would be very much appreciated, as I'm really ...
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0answers
27 views

Continuous convergence [on hold]

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...
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1answer
34 views

Area Between Intersecting Lines - Elegant Solution?

I am running simulations, and the output will be a line y = mx+b. I am interested in the area below the line between x=0 and x=1. I am only interested in the area that is below the diagonal y = x. I ...
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1answer
58 views

integrate this double integral by any method you can. [on hold]

I'm having trouble with this double integral: $$\int_0^2\int_0^{2-x} \exp\left(\frac{x−y}{x+y}\right)\text dy\,\text dx$$
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Strongly continuous semigroup Kolmogorov forward integral equation

Let $\{ P_t \}_{t \geq 0}$ be a SCSF($\mathcal{S}$) (strongly continuous semigroup on $\mathcal{S}$) on the space $(E,\mathcal{E})$, where $E$ is a Polish space, equipped with the ...
3
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1answer
59 views

Integral does not 'converge' despite describing a well-defined area…

I have almost evaluated (where all variables are real including the variable $i$) $$ C_1\int_{a + bt^2}^{i} \frac{r ...
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3answers
79 views

How do I calculate $ \int_{1}^{3} x/(2-x) \;\mathrm{d}x$

$ \int_{1}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $ \int_{1}^{2} \frac{x}{2-x} \;\mathrm{d}x$ + $ \int_{2}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $u = 2-x$ $\lim_{e\to0} \left[ \int_{-e}^{1} \frac{2-u}{u} ...
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2answers
281 views

Calculating the integral $\int_{1/3}^{3}\frac{\arctan(x)}{x^2-x+1}\;\mathrm{d}x$

Can somebody help me calculate the following integral: $$\int\limits_{1/3}^{3}\frac{\arctan(x)}{x^2-x+1}\;\mathrm{d}x$$ I have tried integration by parts, but I got stuck in it. Wolfram also didn't ...
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3answers
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Parametrization of $x^2+y^2-ay=0$

I am to find the circulation of $$y^2 dx + x^2 dy$$ along the (counterclockwise) path $$\Gamma : x^2+y^2-ay = 0$$ both with and without using Green's theorem. Apparently, $\Gamma$ is supposed to ...
2
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3answers
54 views

Which function can be used for Substitution

Find the value of $$I=\int_{0}^{\frac{\pi}{2}}\left(\sin(x)-\cos(x)\right)\,\log(\sin(x))dx$$ Method $(1)$. I splitted up the Integral into two Integrals as $$I=I_1+I_2$$ where ...
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1answer
34 views

Example of a convergent series for which integral test fails?

Is there example of a convergent series for which integral test fails or can not be applied? Just wondering if integral test is the silver bullet of convergence tests, or are there any series that any ...
2
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2answers
36 views

Bound on the integral of a function with multiple zeros

This is a follow-up to this If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that ...
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1answer
53 views

Calculate this double integral [on hold]

Recently took and exam and this was one of the questions and I wanted to check if I did it right Let $R$ be the triangular region in the ($x$,$y$)-plane with vertices $(0,0)$, $(1,0)$ and $(1,2)$. ...
0
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1answer
34 views

Integral with 'reset'

I am trying to mathify the following algorithm description: The algorithm iterates over the elements in the sequence $(f_1, ..., f_n)$, calculating the heuristic function $h(f_k, f_{k+1})$ for ...