# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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 + ...
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### 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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### 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 ...
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### 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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### 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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Let us define $$\boldsymbol{E}(\boldsymbol{x}):=\lim_{\varepsilon\to 0}\int_{D\setminus ... 2answers 58 views ### Solution of integral involving exponential and absolute values [closed] I want to solve this integral \int_0 ^\infty e^{-iwx}e^{-α|x|} dx Any ideas on how to solve it? 0answers 49 views ### Evaluate \int\frac{1}{x}\coth(ax)\sin^2(\frac{xt}{2})dx I am trying to integrate this function: \int_0^\infty\frac{1}{x}\coth(\frac{\hbar x}{2kT})\sin^2(\frac{xt}{2})dx which Wolframalpha (for me) returns nothing, just a blank screen. I thought that it ... 1answer 48 views ### Integration in Calculus and integrals Let f be a function continuous on [0, 1] and twice differentiable on (0, 1). Suppose that f(0) = f(1) = 0 and$$\int_0^1 f(x) \ dx = 0$$Prove that there exists a number x_0 \in (0,1) such ... 1answer 52 views ### What is the relevance of linearity and monotone convergence theorem? From Williams' Probability w/ Martingales: 5.5 is linearity of integration for non-negative \Sigma-measurable functions, and MON is monotone convergence theorem. What are the relevance of ... 1answer 37 views ### Questions about integrals over subsets From Williams' Probability w/ Martingales: What exactly is meant by f|_A? If we have f: \mathbb{R} \to \mathbb{R} and A = [0,1], does that mean f|_A: [0,1] \to \mathbb{R}? Is it that f|_A ... 4answers 217 views ### Using Fatou's Lemmas in proving Scheffe's Lemma Part (ii) Based on Williams' Probability w/ Martingales: Let (S, \Sigma, \mu) be a measure space. Scheffe's Lemma Part (ii): Suppose \{f_n\}_{n \in \mathbb{N}}, f \in \mathscr{L}^1 (S, \Sigma, \mu) and ... 2answers 169 views ### What is a counterexample to the converse of this corollary related to the Dominated Convergence Theorem? Based on Williams' Probability w/ Martingales: Let (S, \Sigma, \mu) be a measure space. Dominated Convergence Theorem: Suppose \{f_n\}_{n \in \mathbb{N}}, f are \Sigma-measurable \forall ... 1answer 29 views ### Question about simple functions Based on Williams' Probability w/ Martingales: Let (S, \Sigma, \mu) be a measure space. A \Sigma-meas function f is called simple if f may be written as a finite sum f = \sum_{k=1}^{m} a_k ... 2answers 85 views ### Why should I take this Contour for I = \int_0^{\infty} \frac{dx}{1+x^3} ? (Analytic Continuation) When discussing analytic continuation, my lecturer used the following example,$$ I = \int_0^{\infty} \frac{dx}{1+x^3} $$I have in my notes that the contour was taken as below. I must admit I was ... 2answers 80 views ### Indefinite integration of \ln(1+x^2)\arctan x We need to evaluate$$\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 \, ...
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 ...
### 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.