All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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23 views

Confused with finding C in economic integral

Question: An automobile company is ready to introduce a new line of cars. They project that the sales will increase by: $P'(t)=10-10e^{-0.1t}, 0\leq t\leq 24$ in t months after the campaign has ...
2
votes
4answers
417 views

How to integrate $\int e^{-x}\arctan(e^x) \, dx$

After trying this multiple ways, I give up. Here's the integral: $$\int e^{-x}\arctan(e^x)\,dx$$ I have set $u=\arctan(e^x)$ and $dv=e^{-x}d\,x$ and have obtained $du=\dfrac{e^x \, dx}{1+e^{2x}}$ ...
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0answers
70 views

negative derivative of integral with time varying domain?

Show that $$\frac{d}{dt}\int_{A(t)}f(x) dx \leq 0 $$ where $f$ is the density function of a zero mean multivariate normal vector with covariance $D=diag(d_1,\ldots,d_p)$ and $A(t)=\{||x+tc||\leq r \ ...
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votes
3answers
169 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
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1answer
82 views

Prove the Jordan lemma i.e. $\int e^{-R\sin{\theta}}< \pi/R$

In complex variables my instructor wrote on the board "Jordan's Lemma", and then, somewhat imprecisely, $$\int e^{-R\sin{\theta}}< \pi/R \;\;\;\; \text{ e.g. } \int \frac{s \sin{x}}{x^2 + 2x + ...
2
votes
2answers
87 views

Showing that a Lebesgue integral tends to zero as $n\to\infty$

I want to show that $\displaystyle\int_{0}^{\infty} \frac{\sin(e^x)}{ 1+nx^2}dx \to 0 $ as $n\to\infty$ from the point of view of Lebesgue integration. Is this as simply as bounding the numerator and ...
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1answer
54 views

Derivative of integral with time varying domain

Let $f:\mathbb{R}^p \rightarrow \mathbb{R}$ be a smooth function. Let $A(t) \subset \mathbb{R}^p$ be varying with time $t$. Is there a nice expression for $$\frac{d}{dt}\int_{A(t)}f(x) dx$$ ?
4
votes
1answer
62 views

How to choose contour in $\mathbb{C}$ to do Residue Integration.

I'm almost sure that there's not any simple way to answer this question, but I'll try. I'm studying complex variables and the method of calculating improper integrals with residues but I'm struggling ...
0
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0answers
103 views

Prove that the determinant of the Jacobian transformation from the interval [-1,1] to any line segment is given by:

This is just for fun...it’s part of a larger numerical application where I’m applying a quadrature rule to a line integral. I’m fairly certain my answer is correct but have no idea how to prove it. ...
7
votes
1answer
198 views

Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

In this post Cleo gives a misterious result containing the following generalized Meijer G-function: ...
2
votes
1answer
121 views

The integral $\int_0^{\infty } \frac{L_m(-x)}{e^{2 \pi x}+1} \, dx$

Could you expain the following sum I seen in a forum $$\int_0^{\infty } \frac{L_m(-x)}{e^{2 \pi x}+1} \, dx=\sum _{n=0}^{\infty } \left(2^{-2 n-1} \left(2^n-1\right) \pi ^{-n-1} \zeta (n+1)\right) ...
1
vote
2answers
53 views

contour integral with arclength

Let |c|< 1 and $\gamma$ be the unit circle , then how do i calculate the following integral? $$\oint_\gamma \frac{|dz|} {{|z-c|^2}}$$ I've tried writing the integral with $d\theta$ but this led ...
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2answers
60 views

Convert double integral from cartesian coordiantes to polar coordiantes

I have the integral $$\int_{-3}^3 \int_0^\sqrt{9-x^2} (x^2 + y^2)^{3/2} {dy}{dx}$$ I cannot solve this in it's current form so I realize that the limit is a circle ${x^2} + {y^2} = 9$ using this I ...
0
votes
3answers
70 views

show that $|\int_\gamma\frac{dz}{z-\frac{3}{2}}|\le4\pi$

if $\gamma:[0,2\pi]\mapsto\Bbb C,\quad \gamma(t)=1+e^{it}$ then show that $|\int_\gamma\frac{dz}{z-\frac{3}{2}}|\le4\pi$ (without computing) I tried : $ |\int_\gamma\frac{dz}{z-\frac{3}{2}}| \le ...
10
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5answers
795 views

Evaluating Double Integral $\int_0^b\int_0^x\sqrt{a^2-x^2-y^2}dy\,dx$

What is the best method for evaluating the following double integral? $$ \int_{0}^{b}\int_{0}^{x}\,\sqrt{\,a^{2} - x^{2} - y^{2}\,}\,\,{\rm d}y\,{\rm d}x\,, \qquad a > \sqrt{\,2\,}\,\,b $$ Is ...
1
vote
1answer
46 views

Power series expansion of a function under the integral sign

I have the following integral: $$S(x)=\int_a^bdtK(x,t)g(t)$$ I don't know the function $g(t)$, but I know it's continuous in the $[a,b]$, so, for the Heine Cantor theorem it's uniformly continuous in ...
1
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2answers
28 views

solving two simple line integrals

First one is : $$\int_\gamma e^zdz,\quad \gamma(t)=\pi ti,\quad t\in[-1,1]$$ my attempt: $z=\gamma(t)=\pi ti \quad dz=\pi idt \quad -1\le t\ \le1, $ then $$\int_\gamma e^zdz=\int_{-1}^1e^{\pi ti}\pi ...
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0answers
58 views

Evaluating $\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h_1^2}dx\int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h_2^2}dx$.

How to evaluate the integral $$\displaystyle\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h^2}dx \int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h^2}dx$$ and ...
3
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3answers
88 views

Orthogonality of Haar wavelet functions

I'm reading about wavelets and I bumped into the follwing: $\text{Haar wavelet is a step function}\; \psi(x), \text{which takes values 1 and -1, when}\; x \;\text{is in the ranges}\; [0, \frac{1}{2}) ...
1
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1answer
267 views

Find the density of the sum of two uniform random variables

Let $X$ and $Y$ be independent and uniform random variables on $(0,1)$. Find the density of $X+Y$. I know $$f_{X+Y}=\int_{-\infty}^{\infty}\,f_X(x)\,f_Y(z-x)dx$$ Now when $x\in(0,1)$, I know ...
4
votes
1answer
80 views

Prove that $\int_0^1 f(x^2)dx\geqslant f\left(\frac{1}{3}\right)$

Let $f:[0,1]\to\mathbb{R}$ be twice differentiable. Suppose $f''(x)\geqslant 0$ for all $x\in[0,1]$. Prove that $$\int_0^1f(x^2)dx\geqslant f\left(\frac{1}{3}\right).$$ I am thinking of using ...
1
vote
0answers
12 views

Mass of function concentrating near origin

Let $$g_t(x)=\dfrac{1}{\sqrt{t}}e^{\frac{-(at+x)^2}{4t}}$$ where $a\in\mathbb{R}$ and $t>0$. Fix $r>0$. Why is it true that $$\lim_{t\rightarrow 0^+}\int_{|y|>r}g_t(y)dy=0?$$ We can show that ...
0
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2answers
38 views

Given an integral, give the given equality?

I've been given this problem These are my attemtps - I'm pretty sure my a and b are wrong. What is the question asking me to do, specifically?
1
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1answer
134 views

Price-Demand, Marginal-price and other financial jargon

So my book likes to assume that I already have a business degree while learning calculus so I need your help to clarify my book's questions. It asks: Price-demand equation. The marginal price for a ...
2
votes
1answer
59 views

Hankel trasformation of acoustic wave equation

We consider a simplified version of acoustic wave equation \begin{equation} \frac{\partial^2 p}{\partial r^2}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^2 p}{\partial z^2}+k^2 ...
1
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1answer
57 views

Prove that a function can be represented as Infinite Darboux sums

$f$ is positive and decreasing monotonously. Both $\int _0^\infty f(x)dx$ and $\sum_{n=1}^\infty f(nk)$ converges. Prove that: $$\lim_{k^+\to 0} k\sum_{n=1}^\infty f(nk)= \int _0^\infty f(x)dx$$ ...
3
votes
1answer
44 views

Differentiating under integral for convolution

I have a function $f\in L^1(\mathbb{R}) $ and $g(x)=\dfrac{1}{2\sqrt{\pi t}}e^{-\frac{(at+x)^2}{4t}}$, where $a,t\in\mathbb{R}$, $t>0$. I want to show that $$\dfrac{d}{dx}\int_{-\infty}^\infty ...
3
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1answer
125 views

Compute $\int_{|z|=1}\frac{\log z}{z}dz$.

Here is a question about contour integration in complex analysis: Compute $$\int_{|z|=1}\frac{\log z}{z}dz$$ I am not sure if I understand the question since the logarithm must be defined in a ...
0
votes
1answer
27 views

Integration Query for solving Bit error rate.

How to solve this integral with complete steps?
0
votes
1answer
82 views

If derivative of a function is non-zero then it is monotone. Since function is monotone, variable can be substituted in integration

I came across this in the text Differential Equations-An Introduction With Applications, by Lothar Collatz: $y'(x)=\frac{dy}{dx}=\frac{f(x)}{g(y)}$ Suppose $f(x)$ is continuous in $[a,b]$ and $g(y)$ ...
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0answers
212 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
1
vote
1answer
1k views

Using trapz and linspace to evaluate an integral in Matlab

I have two functions f(x) and g(x), and I need to find a numerical approximation to the area bounded by the two curves. How do I do this using the trapz and ...
0
votes
0answers
158 views

How to deal with negative x when integrating reciprocal to log

I'm a bit confused as to the right approach to take when integrating $1/x$ to get $\log x$ (for $x \neq 0$). I've seen the following approaches. 1 The approach given in my integration tables, which ...
0
votes
2answers
109 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
1
vote
1answer
62 views

Confused about taking absolute value after integrating reciprocal

I often seem to get caught out when integrating $1/x$ to $\log x$, or similar. Here's an example -- solve $$ \frac{\mathrm{d}z}{\mathrm{d}x} + \frac{1}{2}z = \frac{1}{2}$$ My first attempt was using ...
3
votes
1answer
250 views

Indefinite Integral $\int\frac{3\sin(x)+2\cos(x)}{2\sin(x)+3\cos(x)}dx$ [duplicate]

How can I evaluate this integral? $$\int\frac{3\sin(x)+2\cos(x)}{2\sin(x)+3\cos(x)}dx$$
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0answers
160 views

Definite integral involving exponential, powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} e^{-qx^{ak}(x^2+d^2+2 d x Cos[t])^{-a/2}} dt $$ I am not able to find any related formula. Note that this integral follows ...
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0answers
82 views

Definite integral involving powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} \frac{m}{m+x^{ak}(x^2+d^2+2 d x Cos[t])^{-a/2}} dt $$ I am not able to find any related formula, while I think Mathematica ...
3
votes
1answer
69 views

How to evaulate $\int \cos x \sqrt{5 + \cos^2 x} dx$?

How do I evaluate $$ \int \cos x \sqrt{5 + \cos^2 x} dx? $$
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1answer
40 views

Need help with a certain integrating technique

How do I integrate this? $$\frac{\text{d}P}{\text{d}t} = \frac{(r(t) - B)}{z} \cdot P(t) + c\cdot w$$ The t is just the top limit in the integral. Let me be more specific. I have this: ...
22
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2answers
631 views

Closed form for $\int_0^\infty\frac{\sin x\,\cdot\,\operatorname{Ci}x-\cos x\,\cdot\,\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx$

Is it possible to find a closed form for this integral? $$\mathcal{S}=\int_0^\infty\frac{\sin x\cdot\operatorname{Ci}x-\cos x\cdot\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx,$$ where $\operatorname{Ci}x$ ...
0
votes
0answers
78 views

Integral $\int_{ \ 0}^{\ L} \exp{(\frac{-(x-x_0)^2}{4n})\sin({m\pi\over L}(x-A)}\ \mathrm dx$

How to find integral: $$I=\int_{ \ 0}^{\ L} \exp{\left(\frac{-(x-x_0)^2}{4n}\right)\sin\left({m\pi\over L}(x-A)\right)}dx$$ Thanks in advance. My try: By substitution I get $$z=(x-x_0)^2$$ $$\sqrt ...
2
votes
3answers
138 views

integral of $\,x/\ln(x)$

I'm tried to integrate this integral without any success using integration by parts and substitution. $$\int_0^1\int_{e^y}^e \frac{x}{\ln x} dx dy$$ Does this integral require some other technique?
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0answers
39 views

Integral of convolution difference approaches zero

Let $u(x,t)=f(x)\ast\left(\dfrac{1}{2\sqrt{\pi t}}e^{-\dfrac{(at+x)^2}{4t}}\right)$, and suppose that $f\in L^1$. Show that $$\lim_{t\rightarrow 0^+}\int_{-\infty}^\infty|u(x,t)-f(x)|dx=0$$ How ...
0
votes
1answer
52 views

Solve the following integral

How $$\frac{1}{\pi}\left[\int_{-\pi}^{-\frac{\pi}{2}}\sin2x\cos nxdx+\int_{0}^{\pi}\sin2x\cos nxdx\right]=\frac{-2}{\pi}\frac{1+\cos(\frac{n\pi}{2})}{n^2-4}$$ $n=1,2,3,\ldots$ ?? My Attempt ...
0
votes
1answer
51 views

Integrable functions with sequences of functions (real analysis)

Problem Statement: Let $\phi_n$ be a sequence of nonnegative functions that are Riemann integrable over [-1,1]. Additionally, they satisfy the following properties: $\\$ $$(a) \ \int_{-1}^{1} \! ...
1
vote
1answer
137 views

Power of tangent is odd and positive- integration

So I am currently studying trig substitution and am curious about the process involved in finding an answer. $$ \int \frac {\tan^{3}(x)}{\sqrt {\sec(x)}} dx$$ $$=\int \sec(x)^{-1/2}{\tan} ...
7
votes
3answers
169 views

Why can I make a non-injective variable substitution?

I was using integration by substitution to solve this fairly simple indefinite integral: $$\int xe^{x^2}~dx$$ I simply made the substitution $$x^2=t$$ $$dt=2x~dx$$ But it occurred to me that I don't ...
0
votes
0answers
9 views

Polytime integral approximations in $n$ variables

Let $A = [0, 1]^n$ ($n \approx 16$ for my purposes). I have a function $f$, and I can query $f(x)$ in constant time. I want to computationally approximate $\int_A f(x)$. The natural solution is the ...
2
votes
1answer
39 views

Prove lower bound of integral

I have a continuous function $h:[a,b]\rightarrow\Bbb C$. Let $$M=\sup_{x\in [a,b]}|h(x)|$$ I need to find function $f\in L^2[a,b]$ with: $${||f||}^2=\int_{a}^b|f(x)|^2dx=1$$ such that: $$ ...