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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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1answer
37 views

If $\,f$ and $g$ are continuous with compact support then $f*g$ (convolution ) is also continuous with compact support

I do not know how to find a compact that satisfies $f$ and $g$ support. Could someone explain how find this new compact ? How $f$ and $g$ are continuous in a compact then are bounded in this ...
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2answers
44 views

If $\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$, for $n \in \mathbb{N}$, then $\,\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$

If $\displaystyle\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$ for $n \in \mathbb{N}$. Then prove that $\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$ $$\lambda_n=\int_{0}^{1} \frac{dt}{(1+t)^n}= \frac{2^...
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2answers
45 views

Area between the curves of $2\cos(x)$ and $x/2$

I'm trying to obtain the area between the curve of these two functions (for $x>0$), lets call them $f(x)=2\cos(x)$ and $g(x)=x/2$ and my idea is to get the area under the curve of $f(x)$, then ...
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1answer
32 views

How to find the intersection point between 2cosx and x/2

I'm trying to find the solution to this because I need to find the area between the curves, but I need this intersection point to properly subtract the unnecessary parts. I know how to do it with ...
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1answer
30 views

Substitution for a double integral

Show that $$ 4\int_0^1 \int_0^1 \sqrt{1+x^2+y^2} \, dy \, dx = \frac{8}{3} \int_0^{\pi /4} (1+ \sec^2 \theta )^{1.5} \, d\theta - \frac{2\pi}{3} $$ I have tried letting $1+x^2 = \lambda^2$ , $y=\...
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3answers
2k views

Limit of a sequence of integrals involving continued fractions

The following question was asked in a calculus exam in UNI, a Peruvian university. It is meant to be for freshman calculus students. Find $\lim_{n \to \infty} A_n $ if $$ A_1 = \int\limits_0^...
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3answers
60 views

Seeking methods to solve $\int_{0}^{\infty} \frac{x - \sin(x)}{x^3\left(x^2 + 4\right)} \:dx$

I'm looking for methods to solve the following integral: $$I = \int_{0}^{\infty} \frac{x - \sin(x)}{x^3\left(x^2 + 4\right)} \:dx$$
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0answers
17 views

Finding probability of a random variable dependent on other random variables

Let H,C G, D be three independent random variables where H belongs to Gamma distribution and C follows an exponential distribution, G and D belong from similar distributions with different mean. Rest ...
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15 views

How can I solve the following integrals? [duplicate]

I'm trying to solve these integrals. $\int_R max(x,y) \ dxdy$ $\int_R max(x^2,y) \ dxdy$ where $R=[0,1]^2.$ My idea was splitting $R$ into two disjoint sets, $R1=${$(x,y): 0 \leq ...
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2answers
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Evaluate $I_n = \int_0^{\pi / 2} \sin n \theta \cos \theta \,d\theta$ by integrating by parts twice

By integrating by parts twice, show that $I_n$, as defined below for integers $n > 1$, has the value shown. $$I_n = \int_0^{\pi / 2} \sin n \theta \cos \theta \,d\theta = \frac{n-\sin(\frac{\pi ...
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2answers
44 views

What's this definite integral?

I'm confused about solving this definite integral: $\int_{\pi/6}^{7\pi/6} \sec{x}\tan{x}{dx}$. When I solve it via fundamental theorem of calculus, it's pretty easy to see that the $\sec{x}\tan{x}$ ...
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1answer
76 views

Integral $\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$

I was playing around with some integrals and noticed that some integrals of the form: $$I(a,b,c)=\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$$ Does have a closed form. I am trying to find for what ...
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2answers
47 views

Trying to proof an integral inequality

Let $f_s,f_l:(0,\infty) \mapsto {\mathbb R}$ be monotonic increasing functions with $f_s(x) \leq f_l(x)$ and let $x_s,x_l$ be the smallest positive roots of the equation $1-f_{s,l}(x)=0$ respectively (...
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1answer
48 views

parametrise $x^2+y^2+(x+y)^2=4$

I am trying to evaluate the double integral $$ \iint_D 3 \, dA $$ , where $D=\{(x,y)\in \mathbb{R^2}: x^2+y^2+(x+y)^2\le 4\}$. I need help with a suitable parametrisation for this.
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1answer
16 views

Triple integrals (Find volume): The solid bounded by the sphere $r = 2 cos$ $ \phi $ and the hemisphere $r = 1$, $z$ $\ge$ $0$

Here is the exact question: https://imgur.com/a/cBQC8su! My particular question regards the range of $\phi$; $\phi$ certainly lives between $0$ $\le$ $\phi$ $\le$ $\frac {\pi}{2}$. $\rho = 1$ ...
5
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2answers
105 views

Finding the definite integral $\int_1^e \frac{dx}{x\sqrt{1+\ln^2x}}$

So I have the following problem: $$\int_1^{e} \frac{1}{x\sqrt{1+\ln^2x}}dx $$ Can somebody comfirm that the integral of this is $$\ln|\sqrt{1+\ln^2x}+ \ln x|+C$$ and I that the anwser is $$\ln |\...
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0answers
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Integral involving the Associated Laguerre polynomials

I'm trying to solve this integral $\int_{0}^{\infty} L^n_p L^n_{p'} e^{-x} x^{n-1} dx = \dfrac{1}{n} \dfrac{(p!)^3}{(p-n)!} \delta_{pp'}$ I started with integration by parts where $u = $ $L^n_p L^...
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2answers
37 views

Evaluating a simple integral with the Cauchy residue theorem and a semicircular contour

I am taking a course next week that requires some basic integral techniques from complex analysis and I've been trying to quickly teach it to myself. I was given this sample problem to test my ...
2
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1answer
69 views

Calculate $\int_{a}^{b}\ln x ~dx$ using the definition of integral

I am being asked to calculate the integral $\int_{a}^{b}\ln x~dx$ using the definition of integral (i.e. expressing it as limit of Riemann sums) Here's what I did: Let's divide the interval $[a,b]$ ...
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0answers
15 views

how to break limits of an integral of joint variables [on hold]

if x1 & x2 are uniform random variables then find pdf of Z= X1+ X2 f(x1,x2) = 2,, where 0 <- x1 <- x2 <- 1 BY using CDF method find pdf for Z
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3answers
57 views

Showing $\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_0^1 \mid F^{-1}(u)-G^{-1}(u)\mid du$ with $F$, $G$ CDF functions

Let's $X$ and $Y$ have CDF functions admitting moment of order $1$. Let's be $F$ cdf of $X$ and $G$ cdf of $Y$. I want to show that $$\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_{0}^{1} \mid F^{-1}(...
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1answer
52 views

Evaluating $\int_0^1\frac{\ln(1+x-x^2)}xdx$ without using poly logs.

Evaluate $$I=\int_0^1\frac{\ln(1+x-x^2)}xdx$$ without using polylog functions. This integral can be easily solved by factorizing $1+x-x^2$ and using the values of dilogarithm at some special points....
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0answers
29 views

multiple integral of product of logarithmic function

I am trying to solve an integral of the form \begin{align*} \int\limits_{0<t_0<t_1<\cdots<t_r<1} \frac{\log^{m_0}(1-t_0)\left\{\prod\limits_{j=1}^r\log^{m_j}\left(\frac{1-t_{j-1}}{1-t_{...
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1answer
97 views

Let $a_k\gt 0$ and $a_0\gt \sum_{k=1}^n a_k$ . Show that $\int_0^{\infty} \prod_{k=0}^n \frac {\sin (a_k x)}{x} dx=\frac {\pi}{2}\prod_{k=1}^n a_k$

Let $a_k\gt 0$ and $a_0\gt \sum_{k=1}^n a_k$ . Show that $$\int_0^{\infty} \prod_{k=0}^n \frac {\sin (a_k x)}{x} dx=\frac {\pi}{2}\prod_{k=1}^n a_k$$ I saw this question on the internet somewhere a ...
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1answer
26 views

Is the function in the $L^2$-sobolev space $H^{\frac{1}{2}} \ [- \pi, \pi]$ of order 1?

My professor and as a consequence the rest of my class are saying that the specified function is not in $L^2$-sobolev space $H^{\frac{1}{2}} \ [- \pi, \pi]$ of order 1. TLDR: Just skip ahead to $(\...
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0answers
56 views

Integrating triple product of Bessel functions over a finite domain

Just in case there's any way to simplify this integral or at least transform it to something which is easy to integrate numerically: $$I(a,b,c)=\int_0^1 r J_0(ar)J_0(br)J_0(cr)dr$$ I'm interested in ...
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0answers
28 views

Solution for $\int_0^\infty e^{-(ct)^\alpha} \cos(x t) dt$

I'm trying to evaluate the integral: $\int_0^\infty e^{-(ct)^\alpha} \cos(x t) dt$, where $\alpha$ and $c$ are parameters. This integral arises from trying to solve for the probability density for a ...
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0answers
31 views

Integral $\int\limits_0^\infty \frac{\text{d}q} \cos(qx) (\omega^2-q^2)^{N-1} e^{-s(\omega^2-q^2)^N}$

In my research I am interested in finding the following expression: $$ f{}_\omega^N(x) := \int\limits_0^{\ell^{\,2N}} \text{d}s ~ \mathcal{I}{}_\omega^N(x,s) $$ where the main problem lies in finding ...
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0answers
34 views

Integral of squared Hypergeometric Function

I am trying to integrate the following $\int_{0}^{1} {_2{F}_1}\big(-n,1+2m+n,1+m,1-z\big)^2 dz$, where $m\in\Bbb Z$ and $n\in\Bbb Z$ with $m>0$, $n\geq 0$. (Basically I want to normalise the ...
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0answers
28 views

Show that $\int^{\pi/2}_0\sin^{2n+1}{x} \ dx = \frac{(2)(4)(6)\dots(2n)}{(3)(5)(7)\dots(2n+1)}$ [duplicate]

Show that $$\int^{\pi/2}_0\sin^{2n+1}{x} \ dx = \frac{(2)(4)(6)\dots(2n)}{(3)(5)(7)\dots(2n+1)}$$ How should I approach the question? Do I use Riemann Sum?
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3answers
163 views

Evaluating the integral $\int_0^1 \frac{\cos bx}{\sqrt{x^2+s^2} }dx$

I'd really love to evaluate this integral exactly in terms of known functions, because for large $b$ it becomes a pain numerically. $$I(b,s)=\int_0^1 \frac{\cos bx}{\sqrt{x^2+s^2} }dx$$ Didn't get ...
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3answers
305 views

How to compute $(-1)^{n+1}n!(1-e\sum_{k=0}^n\frac{(-1)^k}{k!})$?

I was doing some work on the integral $\int_0^1 x^ne^x dx$ and I eventually came to this expression in terms of n $$\int_0^1 x^ne^x dx=(-1)^{n+1}n!\biggl(1-e\sum_{k=0}^n\frac{(-1)^k}{k!}\biggr)$$ Now ...
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1answer
24 views

Inner Product, Definite Integral

Does the map $<f, g>$ $=$ $\int _0^1\:\left(\left(f\left(x\right)-\frac{d}{dx}f\left(x\right)\right)\left(g\left(x\right)-\frac{d}{dx}g\left(x\right)\right)\right)dx$ define an inner product on ...
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Tips for Multivariable Calculus [on hold]

I have a final exam for multivariable calculus coming up and I ask for some tips on these topics: finding the bounds on a triple integration problem (especially phi in spherical) proving a limit ...
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0answers
56 views

Proving a definite integral is finite

I have a integral which I have to prove is finite. $$\int_{-\pi }^{\pi } \left(\frac{x \cos x-\sin x}{x^2}\right)^2 dx $$ call the function inside $g(x)$, where $g(x) = (f'(x))^2$ and where $f(x) =...
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1answer
42 views

Definite integrals : how do we approach in solving a problem

While practicing definite integrals I came across a question and now I am stuck Question: let f be a continous satisfying $f(x+y) = f(x) + f(y) + f(x)\cdot f(y)$ for all real $x$ and $y$ and $f'(0)=...
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0answers
27 views

Find the analytic form of expression for the below integral

$$ \int_{0}^{\infty} \frac{1}{a \hspace{0.05cm} e^{br} + b \hspace{0.05cm} e^{ar} + c \hspace{0.05cm} e^{(a+b-c)r} + d \hspace{0.05cm} e^{(a+b-d)r}} dr \hspace{0.1cm}; \hspace{0.9cm} ...
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2answers
95 views

Integral of $\int_0^{\infty} \frac{\sin^2(x)}{x^2+1}dx$ using Feynman integration.

Using $$I(t) = \int_0^\infty \frac{\sin^2(tx)}{x^2+1}dx$$ I want to know how to get an answer using Feynman integration and the Laplace transform of a differential equation. The correct answer is $\...
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0answers
10 views

Inner product between Gaussian radial basis functions

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}$ be the Gaussian radial basis function: $$\phi(x) = \exp(-|x|^2)$$ Let $$f_i(x) = \phi{\left(\frac{x - \mu_i}{\sigma_i}\right)}$$ I'm computing the ...
0
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2answers
78 views

Evaluate the integral $\int_{0}^{1} \frac{\\3^x}{x^2}\mathrm dx $

I don't know how to evaluate this integral $$\int_{0}^{1} \frac{\\3^x}{x^2}\mathrm dx $$ Could you give me advice on how I can do it because I am actually puzzled? I tried to integrate by parts, ...
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1answer
73 views

“Funny Integral” over the Cantor Set

I was thinking about integrals and how one might generalize them to be able to integrate over fractals rather than just over intervals. For example, consider the cantor set $C$. Let us assume that $$\...
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1answer
25 views

Calculating limits of integrals of volume

I have a few problems to do with finding the volume under surfaces. The first ones were fairly simple: $$h(x,y)=xy^2\,,0\le x\le1\,,1\le y\le2$$ $$\therefore V=\int_1^2\int_0^1\int_0^{xy^2}dzdxdy=\...
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votes
1answer
38 views

Evaluate $\int^{2}_{-2} \frac{x^2+x^6\sin{6x}}{x^2 +4} dx$

Solve $$\int^{2}_{-2} \frac{x^2+x^6\sin{6x}}{x^2 +4} dx$$ What I did was $$\int^{2}_{-2} \frac{x^2+x^6\sin{6x}}{x^2 +4} dx$$ $$= \int^{2}_{-2} \frac{x^2}{x^2 +4} + \frac{x^6\sin{6x}}{x^2 +4} dx$$ $$...
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2answers
36 views

find $P(X>Y)$ in $f(x; y) = \frac67(x^2 − y^2) $; $x > 0$; $ y < 1$

find $P(X>Y)$ in $f(x; y) = \frac67(x^2 − y^2) $; $x > 0$; $ y < 1$ I am aware this can be solved with a double integral, such that: $$ P(X>Y)=\int\int\frac67(x^2 − y^2) dydx$$ ...
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2answers
37 views

Questions about Indefinite Integrals and U - Substitution

Let me begin with an example. If we were to integrate the indefinite integral of $(2x)^2$ with respect to $x$ with u-substitution, we would first say that $u=2x$ and therefore $du=2dx$. In order to ...
5
votes
8answers
304 views

How to evaluate $\int_{-\infty}^{\infty}dx \frac{x^2 e^x}{(e^x+1)^2}$

My physics textbook makes use of the result: $$\int_{-\infty}^{\infty}dx \dfrac{x^2 e^x}{(e^x+1)^2} = \dfrac{\pi^2}{3}$$ I'm really curious on how I can derive this but I honestly don't know what to ...
0
votes
1answer
42 views

find the value of $\int_{-a}^{a} \frac{f(x)} {1+e^x} dx $?

Let $a$ be a postive real number. If $f$ is a continious and even function defined on the interval $[-a,a]$, then find the value of $$\int_{-a}^{a} \frac{f(x)} {1+e^x} dx. $$ My answer ...
0
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0answers
11 views

Where is symmetry used in the rewriting of the following integral?

Let $F$ be a CDF that is symmetric about $t^* \in T$. Consider the following integral: $$\int_{t^*-\overline{\delta}}^{t^*+\overline{\delta}}cos(t)f(t)dt$$ Using change of variable and the symmetry ...
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0answers
18 views

Laplace Transform of $f(x) \theta(x) \star f(x) \theta(-x)$

Is there a simplified result for Laplace Transform of $f(x) \theta(x) \star f(x) \theta(-x)$, where $f(x)$ is an even function, $\theta(x)$ is Heaviside function and $\star$ is convolution? For $f(x) ...
1
vote
3answers
62 views

Integral of $e^{-i\theta}e^{e^{i\theta}} d\theta$ over the unit circle

I need to evaluate $\int_{0}^{2\pi} e^{-i\theta}e^{e^{i\theta}} d\theta$. My attempt at a solution was to try and find the antiderivative of $e^{-i\theta}e^{e^{i\theta}}$ and use the antiderivative ...