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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18 views

Integration by substitution—total differential swaps

I'm familiar with the elementary integration by substitution. For the following integral use $u = x+1$, so $\textrm{d}u = \textrm{d}x$ $$\int_b^a\textrm{d}x\,(x+1)\,f(x+1) = ...
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40 views

Integrals of a nonnegative, symmetric, periodic, continuous function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be such that $f$ is nonnegative, symmetric, $\ell$-periodic in both variables, zero exactly on the diagonal of its domain, and continuous. Specifically, $f$ has the ...
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4answers
171 views

Geometric proof that $\int_a^c (x-a)(x-b)(x-c)\ dx=0$ if and only if $b$ is the midpoint of $a$ and $c$.

Let $a<b<c$ be three real numbers, and $f(x)=(x-a)(x-b)(x-c)$. We want to show that $\int_a^c f(x)\ dx=0$ if and only if $b=\frac{a+c}{2}$. The first move is to horizontally shift $f$ by $b$ ...
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0answers
38 views

Volume of Sliced Parallelpiped - Triple Integrals (HW Problem)

I'm trying to use a triple integral to compute the volume of a parallelepiped, which is basically a parallelogram in 3-d. The question goes as follows: Use a triple integral to compute the volume of ...
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1answer
27 views

$f(x)$ is an odd function on $[\frac{-T}{2},\frac{T}{2}]$ and has a period $T$. Prove that $\int_a^x f(t)dt$ is periodic with period $T$.

$f(x)$ is an odd function on $[\frac{-T}{2},\frac{T}{2}]$ and has a period $T$. Prove that $\int_a^x f(t)dt$ is periodic with period $T$. The problem boils down to proving $\int_x^{x+T} f(z)dz = ...
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2answers
129 views

Using the Eulerian integrals evaluate $\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x$

The question asks to evaluate the integral: $$\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x.$$ I have tried a few substitutions but am not getting anywhere. Thanks in advance!
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2answers
29 views

Is $\int z^n e^{az}dz $ a combination of exponentials and polynomials?

We have $$I(n)=\int z^n e^{az}dz=\int z^n \left (\frac{1}{a}e^{az}\right )'dz=\frac{1}{a}z^ne^{az}-\frac{1}{a}\int nz^{n-1}e^{az}dz \\ \Rightarrow I(n)=\frac{1}{a}z^ne^{az}-\frac{1}{a}nI(n-1) \ \ ...
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3answers
58 views

Integrate $\frac{x}{\sqrt{1+x^2}}$

I apologize if this is trivial, but I haven't had to do integral calculus in a while and I can't for the life of me remember how to find the indefinite integral $$\int{\frac{x}{\sqrt{1+x^2}}}\,dx ...
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17 views

Evaluate a certain volume integral

Evaluate $\iiint_B zdV $where B is bounded above by the sphere $x^2 + y^2 + z^2 = 17$ and below by $z = 1$ What I have done: By converting to cylindrical coordinates, I find that $ z = \sqrt{17 - ...
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2answers
40 views

Integral of absolute value: $\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$

I am stuck trying to integrate $$\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$$ Incidentally, I'm interested in solving equation (5) in this paper using the Laplace distribution. I just got ...
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0answers
12 views

$\int_{\mathbb{R}^n} |\log(|f_k|)|dx \to 0$ as $n \to \infty$. Show that $\exists f_{n_i}\to 1 $ a.e . $f_{n_i}$ a subsequence

$\int_{\mathbb{R}^n} |\log(|f_k|)|dx \to 0$ as $n \to \infty$. Show that $\exists f_{n_i}\to 1 $ a.e . $f_{n_i}$ a subsequence. My idea is to use the Riesz Theorem. That is If $f_n → f$ in measure, ...
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3answers
60 views

Simple integral: $\int \frac {1+r}{-r^2+r-1} dr$.

I was solving a much longer exercise, and while solving an ODE, I got this integral $$\int \frac {1+r}{-r^2+r-1} dr$$ I think this must be pretty simple, but I couldn't solve it, my substitutions ...
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1answer
197 views

Finding a cone's surface area using strictly spherical coordinates

I have successfully found the surface area of a cone (without the base) using integration and cylindrical coordinates. Next, I was asked to find it using spherical coordinates and the following ...
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2answers
174 views

Evaluate the integral by changing to spherical coordinates.

$$\int_{0}^{6} \int_0^{\sqrt{36-x^2}} \int_{\sqrt{x^2+y^2}}^\sqrt{72-x^2-y^2} xy~ dzdydx $$ I tried converting it and I ended up with ...
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1answer
33 views

How do I solve triple integrals when given no function?

I got this assignment: It is set, that: 0 < b <= c Now, I have to calculate the volume of the figure given by: x >= 0, y >= 0, x + y <= b and x + y <= z <= c I learnt that I ...
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1answer
47 views

What can we say about $\int_{-\infty }^{\infty }\phi \left( Ax\right) \frac{\phi \left( x\right) }{\Phi \left( x\right) }dx$?

Let $$\phi \left( x\right) =\frac{1}{\sqrt{2\pi }}e^{-\frac{x^{2}}{2}}\text{ and } \Phi \left( x\right) =\int_{-\infty }^{x}\phi \left( t\right) dt$$ be the pdf and cdf of the standard normal ...
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0answers
84 views

Contour Integral of Square root Function. Branch Cuts

I am doing a physics problem and have come across a contour integral that I just don't know how to solve. I do not have the complex analysis background and I am wondering if anyone can explain how to ...
3
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1answer
47 views

Integrating $f(x,y)= \frac{1}{2 \pi (t-s)s} e^{- \frac{(y-x)^2}{2(t-s)}- \frac{x^2}{2s}}$

I want to integrate the function $$f(x,y)= \frac{1}{2 \pi (t-s)s} e^{- \frac{(y-x)^2}{2(t-s)}- \frac{x^2}{2s}}$$ on $[0,\infty)^2$ or in other words. I am looking for $$\int_{0}^{\infty} ...
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1answer
50 views

Prove that g is equal to a constant a.e. [duplicate]

Suppose $g : [0,1] \to \mathbb R$ is bounded and measurable and $$\int_{0}^{1} f(x)g(x) dx=0$$ whenever $f$ is continuous and $$\int_{0}^{1} f(x) dx=0$$. Prove that $g$ is equal to a constant a.e. I ...
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1answer
41 views

Improper Riemann integral versus Lebesgue

What is the relationship between the improper Riemann-integral and Lebesgue integrability? There's a very sloppy section of it in my book with about 4-5 known mistakes found so far, so I fear I have ...
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1answer
38 views

Finding pdf of X+Y - Finan 42.2

I am looking through Marcel Finan's 'A Probability Course for the Actuaries' and I am stuck on problem 42.2. It is as follows: Let X be an exponential random variable with parameter $\lambda$ and ...
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1answer
48 views

Riemann integration of a power function

I'm trying to prove $$ \int_1^\infty x^{-a}dx =\frac{1}{a-1} $$ for $a > 1$ using the definition of the Riemann integral (i.e. without appealing to the Fundamental Theorem of Calculus). To do so, I ...
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1answer
69 views

Two similar integration about continued fractions

Prove that \begin{align*} \int_0^{+\infty} \cfrac{\sin nx}{x + \cfrac{1}{x + \cfrac{2}{x + \cfrac{3}{x + \cdots}}}} \, dx &= \cfrac{\sqrt {\cfrac{\pi }{2}} }{n + \cfrac{1}{n + \cfrac{2}{n + ...
3
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1answer
82 views

Double integral involving beta functions

I want to prove the following result: $$\int_0^1 \int_0^1 {f(xy)(1-x)^{p-1}y^p(1-y)^{q-1}} \mathrm{d}x \, \mathrm{d}y=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)} \int_0^1 {f(t)(1-t)^{p+q-1}} ...
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1answer
33 views

Compute the following by using Cauchy's Integral Theorem

$$\int_0^{2\pi} \frac{dt}{1+a^2\cos{t}}$$ where $ 0<a<1$. Could someone please help me? I know that I am supposed to do a change of variables, but at this point, I cannot even think straight.
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1answer
28 views

About the convergence of an integral and a double series

Let $k>0$ a real number and $N>0$ a natural number. For a work, I need to prove the convergence of ...
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3answers
43 views

At what point to the curves $y=ax$, $y=a/x$, and $y=x^3/a$ Intersect with one another when a>0

Hi this has been bugging me for a while now I have the answers however they only detail the points I have so far been unable to figure out how to find these points. Image of problem with arrows to ...
3
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1answer
121 views

Evaluation of the following integral

How do I evaluate the following integral? $$I=\int_{0}^{\infty}\frac{\sinh(a)k\ dk}{\cosh(k) + \cosh(a)}, \qquad a \geq0$$
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1answer
19 views

A definite integral with exp and cos^n

I came across the following integral formula, left as an exercise, in Mark Kac's "Statistical Independence in Probability, Analysis, and Number Theory." I've been unable to prove it and would like a ...
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2answers
56 views

Value of the integration $L=\int_{0}^{1}\frac{1}{1+x^8} dx$?

Given $$L=\int_{0}^{1}\dfrac{1}{1+x^8} dx$$ Then, $L<1$ $L>1$ $L<\pi/4$ $L>\pi/4$ Which options are true. I concluded that $1$ is true as $\frac{1}{1+x^8}\leq 1$. But I am not able to ...
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1answer
35 views

Relating $dS$ and $d\theta$ for computation of line integrals

I'm asked to compute $$\int_C \vec{F} \cdot d\vec{s}$$ where $\vec{F} = A_0(x\hat{y} - y\hat{x})$, along a circular path, counterclockwise, about the origin with radius 4. I begin by writing $$ ...
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1answer
24 views

Use the fundamental theorem to evaluate the line integral?

For part a), I found $||r'(t)|| = \sqrt{18}(t^2+1)$ For part b), I solved $\int_{-1}^{1}f(t)*||r'(t)|| dt = \sqrt{2}\frac{24}{5}$ But I am unsure how to solve part c). I know that the path ...
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0answers
15 views

What is the integral of $\phi(t)$ where $\phi(t)$ = $2\pi f_k*\int_{-\infty}^{t}acos(2\pi f_m t)dt$?

Take $f_k, f_m, a$ to be constants I'm following this calculation from an example from a book. It becomes easier with a substitution $u = 2\pi f_m t$ but then the integral needs to be evaluated at ...
2
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1answer
76 views

How do I evaluate the following Integral

The integral is $$\int_{0}^{\infty}dx\frac{1}{\sqrt{1+a(1+x^2)^m+b(1+x^2)^{m-2}x^2}},$$ where $m, a$ and $b$ are real numbers such that the integral is definitely convergent. Any ideas on how to solve ...
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0answers
31 views

Computing the Lebesgue integral of a power function

I'm studying a book on integration (Lebesgue Integration on Euclidean Space by Frank Jones) and am having trouble with the following problem: Define the open set $$ G = \{(x,y)\in \mathbb{R}^2: 1 ...
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1answer
24 views

Estimating two integrals over the sphere

Is there any easy way to estimate the following integrals over the uniform n-dimensional ball $B = \{\vec{x} : \left\|\vec{x}\right\|^2 \leq 1\}$? $$ \begin{split} f(c) &= \int_B e^{-cx_1} dx\\ ...
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1answer
93 views

Double integration in polar coordinates between two circles

I am trying to integrate converting to polar coordinates, between two circles. $$A = \iint_D x \,\mathrm{d}A $$ Ant the domain of integration is set to be the region in the first quadrant between ...
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2answers
155 views

Continuity of a certain vector field

Let us define $$\boldsymbol{E}(\boldsymbol{x}):=\lim_{\varepsilon\to 0}\int_{D\setminus ...
2
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2answers
58 views

Solution of integral involving exponential and absolute values [closed]

I want to solve this integral $\int_0 ^\infty e^{-iwx}e^{-α|x|} dx$ Any ideas on how to solve it?
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0answers
49 views

Evaluate $\int\frac{1}{x}\coth(ax)\sin^2(\frac{xt}{2})dx$

I am trying to integrate this function: $\int_0^\infty\frac{1}{x}\coth(\frac{\hbar x}{2kT})\sin^2(\frac{xt}{2})dx$ which Wolframalpha (for me) returns nothing, just a blank screen. I thought that it ...
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1answer
48 views

Integration in Calculus and integrals

Let $f$ be a function continuous on $[0, 1]$ and twice differentiable on $(0, 1)$. Suppose that $f(0) = f(1) = 0$ and $$\int_0^1 f(x) \ dx = 0$$ Prove that there exists a number $x_0 \in (0,1)$ such ...
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1answer
52 views

What is the relevance of linearity and monotone convergence theorem?

From Williams' Probability w/ Martingales: 5.5 is linearity of integration for non-negative $\Sigma$-measurable functions, and MON is monotone convergence theorem. What are the relevance of ...
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1answer
37 views

Questions about integrals over subsets

From Williams' Probability w/ Martingales: What exactly is meant by $f|_A$? If we have $f: \mathbb{R} \to \mathbb{R}$ and A = [0,1], does that mean $f|_A: [0,1] \to \mathbb{R}$? Is it that $f|_A ...
3
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4answers
217 views

Using Fatou's Lemmas in proving Scheffe's Lemma Part (ii)

Based on Williams' Probability w/ Martingales: Let $(S, \Sigma, \mu)$ be a measure space. Scheffe's Lemma Part (ii): Suppose $\{f_n\}_{n \in \mathbb{N}}, f \in \mathscr{L}^1 (S, \Sigma, \mu)$ and ...
3
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2answers
169 views

What is a counterexample to the converse of this corollary related to the Dominated Convergence Theorem?

Based on Williams' Probability w/ Martingales: Let $(S, \Sigma, \mu)$ be a measure space. Dominated Convergence Theorem: Suppose $\{f_n\}_{n \in \mathbb{N}}$, $f$ are $\Sigma$-measurable $\forall ...
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1answer
29 views

Question about simple functions

Based on Williams' Probability w/ Martingales: Let $(S, \Sigma, \mu)$ be a measure space. A $\Sigma$-meas function f is called simple if f may be written as a finite sum $f = \sum_{k=1}^{m} a_k ...
1
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2answers
85 views

Why should I take this Contour for $I = \int_0^{\infty} \frac{dx}{1+x^3} $? (Analytic Continuation)

When discussing analytic continuation, my lecturer used the following example, $$ I = \int_0^{\infty} \frac{dx}{1+x^3} $$ I have in my notes that the contour was taken as below. I must admit I was ...
2
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2answers
80 views

Indefinite integration of $\ln(1+x^2)\arctan x$

We need to evaluate $$\int \ln(1+x^2)\arctan x \, \mathrm{d}x$$ My thoughts were to set $u = \arctan x \implies x = \tan u$ so that our integral is transformed to $$\int 2u \sec^2 u \ln \sec u \, ...
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3answers
34 views

Joint Distribution and Covariance?

Problem: Let X be a random variable such that $X \sim N(0, 1)$ Let W be a random variable independent of X such that $\Pr [W = 1] = \Pr [W = −1] = \frac12$. Define Y = XW Show that X and Y are ...
1
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2answers
55 views

Integration with substitution of trig: $\int\frac1{(x^2-4)^{3/2}}dx$

$$\int\frac1{(x^2-4)^{3/2}}dx $$ I'm unsure how to go about this integral after subbing $x=2\cosh(\theta)$; to the power of $3/2$ is confusing me when trying to simplify. Thanks.