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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-1
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0answers
18 views

How does this “integration by differentiation” method work

Apparently, the integral of a function f(x) from a to b can be done through differentiation through this method: $$ \int_a^b f(x)dx = \lim_{x \rightarrow \ 0 } f(\frac{d}{d x} ...
0
votes
0answers
36 views

Computing the limit $\lim_{n \to \infty} \sum_{k=1}^{n} \sqrt{n^4+k}\sin \frac{2k \pi}{n}$

I'm pretty sure this is a Riemann sum, but I can't solve it. I tried to write it as the integral of a function which equals the product of $\sin(2 \pi x)$ and something, but so far it didn't work. ...
1
vote
1answer
21 views

Riemann integrability of a function $f:[0,1] \to \mathbb{R}$ with exactly 3 points of discontinuity and specific slopes

A function $f: [0,1] \to \mathbb{R}$ has exactly $3$ points of discontinuity is strictly increasing on $[0,\frac{1}{2}]$ and strictly decreasing on $(\frac{1}{2},1]$. $f$ is Riemann integrable Is ...
1
vote
0answers
19 views

Integration of forms on non-simply connected manifolds

What I know is that closed forms are not exact on non-simply connected manifolds, so for instance, if $E$ is a closed form, then $dE = 0$ but $\int_\gamma E \neq 0$, where $\gamma$ is a ...
0
votes
0answers
15 views

Integral of a function depending on two variables

I have a function that depends on two parameters, say $f(x,y)$. Now I need to integrate this function from $0$ to $y_{max}$. I have all the values of $x$ and $y$ and also the values of this function ...
0
votes
3answers
33 views

Compute $\int_0^\infty xe^{-nx} \cos{(mx)} dx$ [on hold]

How to find the improper integral $$\int_0^\infty xe^{-nx} \cos{(mx)} dx$$ Thank you!
0
votes
1answer
46 views

Integral $\int \frac{dx}{x^4+1}$ [duplicate]

Find $$\int \frac{dx}{x^4+1}$$ I found a possible solution to this question here However, I was wondering if there is a "nicer" solution, that would more understandable to a person who isn't ...
2
votes
1answer
45 views

$\int_{a}^{b} f(x) = \int_{a}^{b}A dx - \int_{a}^{b} B dx$

Let $f:[a,b] \rightarrow \mathbb{R}$ be an integrable function. Define A = \begin{cases} f(x), & \mbox{if } f(x) \ge 0\mbox{} \\ 0, & \mbox{if } f(x) < 0\mbox{ } \end{cases} Define B = ...
2
votes
1answer
25 views

Finding the Antiderivative of a complex function

I need to find the antiderivative of $f(z) = z\log(z)$, but I am confused on how exactly to do that. So we need to find $\int z\log(z)dz$ right? Since $z = x+iy$, then $\int (x+iy)\log(x+iy)(dx + ...
4
votes
1answer
41 views

Counter-example to $\int_0^\infty f(x) dx=\lim_{t\to\infty} \int_{1/t}^t f(x) dx$

I want to prove or disprove the statement that, for a function $f$ that is continuous on $(0,\infty)$, we have $\displaystyle{\int_0^\infty f(x)\ dx=\lim_{t\to\infty} \int_{1/t}^t f(x)\ dx}$. My ...
0
votes
2answers
31 views

Complex Integration for $\int te^{\alpha t}\cos(\beta t)dt$

How can I solve the general integral $$\int te^{\alpha t}\cos(\beta t)dt$$ using complex integration? Usually we split the integral into real and imaginary parts, but I don't know how we can do that ...
0
votes
1answer
17 views

Can't solve second order ODE with variation of parameters or undetermined coefficients

I have to solve $$y''+4y' +y=\frac{e^\left(-2x\right)}{x^2}$$ The homogenous equation is easy enough to solve and I got $$y(x) = c_1e^{-2 + \sqrt{3}} + c_2e^{-2 - \sqrt{3}}$$ Doing variation of ...
0
votes
1answer
27 views

Double integral inequality proof

Prove that $$\iint_D \sin^2(x+y)\, dA \le \iint_D \sin(x+y)\, dA$$ where $D= \{(x,y)\, |\, 0\le x+y\le \pi, 0\le y\le \pi\}$. I really don't know where to begin with this problem. I'm not sure if I ...
1
vote
2answers
41 views

Solving 2nd order linear recurrence with non-constant coefficients

I am trying to find a general solution to the following definite integral: $$F_{n}{\left(a,b;z\right)}:=\int_{a}^{z}\frac{x^{n}}{\sqrt{\left(x-a\right)\left(b-x\right)}}\,\mathrm{d}x,\tag{1}$$ ...
0
votes
0answers
23 views

Application of Jensen's Inequality. Correct?

Help would be appreciated. Consider $x \in (0,1)$ and $f(x)=x^2$ which is convex we want to show that $\mathbb{E}\Big[f(X)\Big] \geq f\Big[\mathbb{E}(X)\Big]$. Therefore, $\mathbb{E}\Big[X^{2}\Big] ...
4
votes
2answers
78 views

Different results when integrating 1/(x^2-9) with computer tools

For checking my calculations, I generally use Wolfram Alpha or a similar tool. Today, I wanted to check the following integral: $$\int {\frac{1}{{9 - {x^2}}}} dx$$ From my calculations, using ...
-1
votes
0answers
8 views

Please help with Integral Solving of Hyperbolic Decline: b = -(d/dt)*(1/D) [on hold]

I am trying to derive the Hyperbolic Decline equation of D = [1/(1/Di)+bt] from the integral b = -(d/dt)*(1/D). Anyone able to show me the steps of this? I cannot complete the derivation. Thank you. ...
0
votes
0answers
19 views

Why does the limit of the p-series converge to a non-zero constant?

Here, the limit of the integral apparently converges to infinity as x goes to infinity. What I don't understand though, is how could it possibly converge to a constant of 5/7? If you multiply 5/7 ...
0
votes
0answers
21 views

$\int X dt$ integral of random variable

Define $$\int X dt$$ where X is a continuous uniform random variable that can take on any value (0,1). Also, $\int X dt \not = X \int dt$. In other words, $X$ takes on a different, but still random, ...
1
vote
1answer
26 views

Leibniz integral rule?

Suppose that $a$ and $b$ are fixed constants and let $f(x) = \int_{x'=a}^{x'=b} g(x,x') \, dx'$. If x' is in the same direction as x, why is it true that \begin{equation} \frac{df}{dx} = g(x,x') ...
-4
votes
0answers
18 views

What should I do about this problem after finding the eigenvalues? [on hold]

MATHLAB Proyect, problem #4 Stability of Linear Systems
2
votes
1answer
50 views

Find $\int{\dfrac{x}{\sqrt{x+1}+\sqrt[3]{x+1}}dx}$

I have problem with this integral: $$\int{\dfrac{x}{\sqrt{x+1}+\sqrt[3]{x+1}}dx}$$ My idea was to solve it with substition of $$t=\sqrt[6]{x+1}$$ because ...
0
votes
1answer
24 views

Solving $y'-\frac y x=0$ with integrating factor

How do you solve $y'-\frac y x=0$ the answer should be $y=ex$ but I can't get to that point. I tried using the integrating factor but I can't get it to add up.
3
votes
1answer
36 views

Integration by parts - when we do not know the derivative

I was just a bit unsure of how this exactly goes. So I was wondering if the following expression is correct: $$\int_a^b u(x)v(x)\,dx= \left[ \left( \int_{-\infty}^x u(t)\,dt \right) v(x) ...
0
votes
1answer
19 views

Riemann integrability implies Darboux integrability

I'm trying to prove the following: Suppose that given $\epsilon>0$, there is a $\delta>0$ such that if $P$ is any partition of $Q$ (a rectangle in $\mathbb{R}^n$) of mesh less than $\delta$, ...
1
vote
1answer
37 views

Integrating $2(x+y+z)$ over the volume of $x^2+y^2+4(z-1)^2=4$

Consider the ellipsoid: $$x^2+y^2+4(z-1)^2=4.$$ This surface can be parametrised by: $$\vec{r}(u,v) = 2\sin(u)\cos(v)\vec{i}+2\sin(u)\sin(v)\vec{j}+(1+\cos(u))\vec{k}$$ with $$u \in [0,\pi], \quad v ...
0
votes
3answers
57 views

How to compute this integral? [on hold]

I don't really know where to start with this. $$\int \sqrt{x^2+y^2+1}\quad dx$$
2
votes
2answers
30 views

Evaluating integrals in R^m

Let $|\cdot|_m$ denote the Euclidean norm in $\mathbb{R}^m$. Then I wish to prove that $\displaystyle\int\limits_{\mathbb{R}^m}|x|_me^{-|x|_m}dx<\infty$ It's kinda embarrassing to say this, but ...
0
votes
0answers
33 views

Fractional derivative definition

Suppose that $f(x) \in C^1$ for a $x \in [a, x]$. Then a regularization of Riemann-Louisville fractional derivative is defined as: $ \frac{1}{\Gamma(1-b)} \frac{d}{dx} \int_{a}^{x}\left( ...
1
vote
1answer
24 views

Wronskian second solution of $2(1-x)y''-3y'+\frac{y}{x}=0$

The question had asked to use Wronskian method to show a second solution to the DE $$2(1-x)y''-3y'+\frac{y}{x}=0$$ is $$y_2(x)=1 + y_1(x)\operatorname{atanh}[\sqrt{1-x}]$$ where ...
1
vote
1answer
47 views

Integral of $\frac{4}{x+1}$ from $4$ to $\infty$.

I want to calculate the integral $$\int_4^\infty\frac{4}{x+1}dx.$$ I know that the result is $$\lim_{x\to\infty}(4 \ln (x + 1)- 4 \ln (5)),$$ then I get $\infty - \ln (625)$. Is it still ...
3
votes
0answers
33 views

Maximum value of an integral.

Define $$f(x)=\int_0^1e^{|t-x|}dt$$ I have to find the maximum value of $f(x)$ when $0 \leq x \leq 1$. To remove the modulus, I wrote $$f(x)=\int_0^xe^{x-t}dt + \int_x^1e^{t-x}dt$$ ...
1
vote
2answers
21 views

$(f^{-1})'(w)$ for a biholomorphic $f$

Let $U$ and $V$ be open sets containing $0$ in $\mathbb{C}$ such that $f : U \rightarrow V$ is a biholomorphism. $f (0) = 0$. Then show that for every $r > 0$ such that $D(0, r) \subset U$ there ...
0
votes
0answers
34 views

Expand an integral

An integral looks like: $$ \int f(r,r') g(r') d r' = a(r) g(r) + b(r) \partial_r g(r) + g(r) \partial_r c(r) + \cdots $$ where $f(r,r')$ is a function of $r$ and $r^\prime$ (coordinates) and ...
-3
votes
0answers
66 views

I have simplified my integration to the following integration .and,I got stuck here .Could anybody help me? [on hold]

$$ \int _{0}^{\infty }b^{x}\frac{\Gamma(r-1)}{\Gamma(x)\Gamma(r-x)} dx $$ where r is positive integer and b is ratio of two probabilities.I need to get this integration.
1
vote
2answers
61 views

How to integrate cos3θ/(5−4cosθ) from 0 to 2π?

How do you find the following integral using the theory of Residue? $$\int_{0}^{2\pi}\frac{\cos(3\theta)d\theta}{5-4\cos(\theta)}$$ I've substituted $\cos(\theta)$ and $\cos(3\theta)$ for ...
1
vote
1answer
42 views

Closed form solution of $\int \exp(-a (b-x)^{3/2}-cx)\text dx$

Does following integral have a closed form solution (a, b, and c are constants) $$ \int \exp(-a (b-x)^{3/2}-cx)\text dx $$ If not possible, what about a function with close behavior. $$ \int \exp(-a ...
1
vote
2answers
40 views

Evaluating a definite integral using substitution.

I have to evaluate the definite integral: $$\int_0^2 (e^{4x} - 4x)^4(e^{4x} - 1)dx$$ I am having trouble evaluating this because I am not sure what to put as $u$. I was thinking about putting $u=4x$ ...
1
vote
2answers
95 views

Physically impossible to find the constant

How can we show $$ g(a) = \int _a^{a+1} \left\{x\right\} \cdot \left(1 - \left\{x\right\} \right)\:dx = \mbox{const} $$ where $\{x\}$ is the fractional part of $x$.
0
votes
1answer
38 views

A lower bound for $\left|\int_a^b f(t) dt\right|$.

It is known that, if $f : [a; b] \rightarrow \mathbb R$ is integrable, then $f$ is bounded, $|f|$ is integrable and $$\left|\int_a^b f(t) dt\right|\leq \int_a^b|f(t)|dt$$ My question is the following. ...
2
votes
3answers
66 views

Integration by substitution (Why can't I use $u =\sin^2 x$ for $\int \sin^2 x \cos x \space dx$ )

I know that I have to use $$u=sinx$$ and so we have $$du = cosx dx$$ hence we have $$dx = \frac{du}{cosx}$$ and so we have $$\int u^2cosx \frac{du}{cosx}$$ which is equal to $$\int u^2 \space du$$ ...
3
votes
2answers
63 views

Can anyone help with these integrals?

$$1.\int{x\over \sqrt{1-\sqrt{1-x^2}}}dx$$ $$2.\int{\sqrt{x+1}+2\over (x+1)^2- \sqrt{x+1}}dx$$ $$3.\int{\sqrt[3]{1-\sqrt[4]{x}}\over \sqrt{x}}dx$$ Just need the general idea, im not expecting someone ...
-2
votes
2answers
70 views

Impossible to prove that is unbounded!

How we can demonstrate that $$\int _1^e\:\left(1+\log\left(x\right)\right)^ndx$$ is unbounded as $n\to \infty$, without using Bernoulli's Inequality?
2
votes
1answer
34 views

How do I integrate the inequality $ \frac{f(\frac{1}{2}+h)+f(\frac{1}{2}-h)}{2} \leqslant f(\frac{1}{2})$ over the range $h\in[0,1/2]$?

I would like to know the formal steps and theory. I was told that, by integrating this inequality, I can achieve one of the definitions of a concave function in the interval [0,1]. Thanks for your ...
4
votes
1answer
47 views

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$

How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ? Inspired by ...
0
votes
1answer
51 views

How do we prove such integral? [duplicate]

$$\int_0^\infty\frac 1{1+x^n}dx=\frac\pi n\csc\left(\frac\pi n\right)$$ If we want to prove the left side equal to the right side in this case, how do we start? How do we prove this definite ...
-2
votes
1answer
36 views

Direct comparison test for ( Improper ) Integrals [on hold]

How we can prove with direct comparison test for ( Improper ) Integrals that is bounded: $\int _1^n\:e^{-x^3}dx$ ?
1
vote
2answers
64 views

Inequality very difficult to show

1) $\int _0^1\:\frac{x^n}{x^n+1}dx\ge \int _0^1\:\frac{x^{n+1}}{x^{n+1}+1}dx$ but I dont want to use $I_{n+1}-I_n$ 2) How we can prove with direct comparison test for ( Improper ) Integrals that is ...
5
votes
3answers
56 views

Solving the differential equation $(x^2-y^2)y' - 2xy = 0$.

I am trying to solve the equation $$ (x^2-y^2)y' - 2xy = 0. $$ I have rearranged to get $$ y' = f(x,y) $$ where $$ f(x,y) = \frac{2xy}{x^2-y^2}. $$ From here I tried to use a trick that I learned ...
1
vote
1answer
32 views

Very difficult to prove a convergent with Weierstrass

How we can prove that is monotone and bounded: $I_n=\int _1^n\:e^{-x^3}dx\:$ , Have any ideea how we can solve? and explain all to understand, I am a student... P.S: for all guys on this site, you ...