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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How to integrate $e^{-t^2}$?

Anyone know how to integrate the following? $$ \int_0^{+\infty} \! e^{-t^2} \, \mathrm{d}t $$ Thanks
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5 views

Stieltjes integral with discontinuous integrator

I am asked to solve the following Stieltjes integral: Compute $\int_0^6 f\, dg$, where $f(x) = 6x-x^2$ and $g(x)$ is defined by: $$ g(x) = \left\{ \begin{array}{ll} x^2 &\hbox{for $0\leq x ...
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2answers
34 views

Expanding the integrand

Can anyone help me find the solution to this integral: $$\int\limits{(t-4)(t-2)^{4/5}}dt?$$ I think I need to expand the integrand but I do not know how. Thanks a lot!
4
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3answers
84 views

Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
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0answers
20 views

Covariance of two integrated Brownian motions

I have a question that is similar to the one here: covariance of integral of Brownian, but the answer that I come up with does not match what the book claims the answer is. Given that $$X_t = ...
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3answers
22 views

Calculate $\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}$

I have tried several methods but even I can not calculate. $$\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}\,dx$$ If anyone can help, it was part of a test and still I ...
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0answers
24 views

Help With Limit of Integral

So I am working through some practice problems, and on one of them I can't get the second part: For $x\in(0,\infty)$ and $n\in\{1,2,3,\dots\},$ let $f_n(x)=\frac{e^{\sin\left({x^2/n}\right)}}{1+x}.$ ...
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1answer
11 views

Revolving an unknown equation around the x and y axes

The first quadrant region enclosed by the x-axis and the graph of y = ax - x^2 traces out a solid of the same volume whether it is rotated about the x-axis or the y-axis. What is the value of a?
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1answer
58 views

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

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $$\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}.$$ My attempt: I perform linear substitution $u=x-a+c$ ...
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1answer
31 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
61 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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55 views
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0answers
20 views

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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39 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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0answers
14 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 ...
1
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1answer
46 views

Integration by parts tricks [on hold]

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 ...
3
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3answers
71 views

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
27 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
18 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
19 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, ...
0
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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
34 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
60 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
22 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
67 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
86 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
3
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0answers
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 ...
1
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1answer
29 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 ...
2
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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
45 views

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
237 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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20 views

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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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 ...
1
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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 ...
2
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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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0answers
21 views

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 ...