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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0
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2answers
65 views

Evaluate $\lim_{y\to\\+0} \int_{0}^{1} \exp\left(-\frac{\arctan x}{y}\right)\,\mathrm dx$

I am trying to evaluate the following $$\lim_{y\to\\+0} \int_{0}^{1} \exp\left(-\frac{\arctan x}{y}\right)\,\mathrm dx$$ Seems useful to bring limit under integral, but can't see the solution. ...
0
votes
2answers
151 views

Finding the volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$

The problem is stated below: Let $V$ be volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$. Express $V$ as a multiple integral, whose limits should be clearly stated. Hence ...
2
votes
1answer
190 views

Approximating an integral with another integral with finite limits

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
5
votes
2answers
93 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
1
vote
2answers
77 views

Calculate the mean of the normal distribution function $\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ by integration.

I know that it must be $\mu$ but I cannot get the answer. This is my attempt so far: Normal distribution function = $N(x)=\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ $$\langle ...
2
votes
3answers
123 views

Evaluate $\int_0^\pi \arctan(\cos x)\,\mathrm dx$

I need to Evaluate $$\int_0^\pi \arctan(\cos x)\,\mathrm dx$$ . I tried to make an exchage $t=\cos x$ and then take the integral by parts
3
votes
3answers
99 views

Help with $\int \frac{1}{(\sin x + \cos x)}$ [closed]

Kindly solve this question $$\int \frac{1}{(\sin x + \cos x)} dx$$ I reached up to $$\frac{(1+\tan^2x)}{1-\tan^2x + 2\tan x}$$
2
votes
1answer
65 views

Evaluate $ \int \frac{\tan(x)}{2+\sin(x)}dx $

How do you evalute this integral? $$ \int \frac{\tan(x)}{2+\sin(x)}dx $$
3
votes
3answers
89 views

Evaluate $\int_0^\infty e^{-x^2}\cos\frac{t}{x^2}\,\mathrm dx$

How to integrate $$\int_0^\infty e^{-x^2}\cos\frac{t}{x^2}\,\mathrm dx$$ where $t$ is a constant/parameter?
2
votes
2answers
49 views

Integral of $\dfrac{\cos(x)}{5+3\cos(x)}$

I was doing $$\int\!\mathrm{d}x \dfrac{\cos(x)}{5+3\cos(x)}$$ and using the substitution $\cos(\theta) = \dfrac{1-t^2}{1+t^2},\quad t = \tan\left(\dfrac{\theta}{2}\right)$ ...
0
votes
0answers
28 views

Multiple integral of iterated kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel $$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$ should be calculated. However, it is not ...
1
vote
2answers
35 views

Are these the same? $\int_{-2}^2(2x^2-x)^4$ and $2\int_{0}^2(2x^2-x)^4$

$$\int_{-2}^2(2x^2-x)^4\,\mathrm dx\quad\text {and}\quad2\int_{0}^2(2x^2-x)^4\,\mathrm dx$$ I tried to solve this and got different answers, but the other problems that I did got the same answers. ...
4
votes
1answer
63 views

Doubt in solution for evaluating $\int_0^1\int_0^1\int_0^1(1+u^2+v^2+w^2)^{-2}du~dv~dw$.

I've a doubt in the answer for evaluating the following integral: $$\int_0^1\int_0^1\int_0^1(1+u^2+v^2+w^2)^{-2}du~dv~dw$$ solution: call this integral as $I$ .By symmetry we may compute it ...
0
votes
0answers
25 views

Approximate the inverse Laplace transform

I am struggling with an inverse Laplace transform for a long time! Assume we have a function $m(t)$ and its Laplace transform is denoted by $M(s)$. I have derived the expression of $M(s)$ by some ...
0
votes
0answers
32 views

Integral and limits

My lecturer made the following remark: $$\frac{1}{2\pi i}\int_{\sigma_{0}-i\infty}^{\sigma_{0}+i\infty} y^{s} \frac{ds}{s} \neq \lim_{T\rightarrow \infty} \frac{1}{2\pi ...
2
votes
1answer
28 views

Doubt on integrating $f$ on a given region .

I was asked to integrate the following function: $f(x,y,z)=1-z^2$ on $U$ where $U$ is a pyramid with the top vertex $(0,0,1)$ and base vertices :$(0,0,0),(1,0,0),(0,1,0), (1,1,0)$ the ...
3
votes
1answer
36 views

Is the following function Riemann integrable?

Consider $f:\mathbb{R^2}\to\mathbb{R}$ defined by: $$f(x,y):=\arctan\frac{1}{x-y}\quad\forall x\neq y$$ $$f(x,x):=0$$ Is the function Riemann integrable in the square $[0,1]\times[0,1]$? I just ...
1
vote
1answer
68 views

Find expected present value of a continuous payment stream

I have a question for the financial part of my course which I am struggling to answer as i am not sure my answer makes sense. Question: Time is counted from the present t = 0 in years. Suppose for ...
2
votes
1answer
74 views

use fundamental theorem of calculus to find a function $f(x)$ and a number $a$

I thought I understood the fundamental theorem of calculus but I'm confused on the following problem.. Use the Fundamental Theorem of Calculus to find a function $f(x)$ and a number $a$ so that ...
0
votes
1answer
42 views

Expectation of random varible with normal distribution composed with exponential [duplicate]

I am trying to find $\mathbb{E}(e^{-X})$ where $X$ is a random variable with a general normal distribution. I end up with $$(2\pi \sigma)^{-\frac{1}{2}} \int_{-\infty}^{\infty} ...
5
votes
1answer
188 views

Proof that $\int_{0}^{1}\frac{dx}{1+x^6}=\frac{\pi+\sqrt3\log(2+\sqrt3)}{6}$ without residues.

How do you prove that $$\int_{0}^{1}\dfrac{dx}{1+x^6}=\frac{\pi+\sqrt3\log(2+\sqrt3)}{6}$$ My steps: First sub $\displaystyle u=x^3, \sqrt[3]u=x, dx=\dfrac{u^{-2/3}}{3} ...
0
votes
1answer
44 views

Integration over a spherical surface

Suppose that we have let's say a function of "something" that is everywhere on the spherical surface zero except at one point it is finite. Why is the integral of such a function over the surface is ...
0
votes
1answer
36 views

How can we find another path

Can you help me about this question please, Thank you..
0
votes
1answer
39 views

Relation between cone radius and distance from cone apex

I am stuck on the following problem; Let a cone with height $h$ and base area $B$ have the density $\rho (x) = \rho_{0} \frac{3x^{2} + 2xh}{h^2}, 0 \leq x \leq h$ Where $x$ is the distance from ...
0
votes
0answers
49 views

Numerical integration formula

I'm looking for a numerical method/scheme which can be used to solve the following equation $$ \int_{t_n}^{t_n+h} \sin\left((t_n+h-s)\omega\right) g\left(x(s)\right)ds = \frac{1}{2\omega} \sin^2(h ...
14
votes
9answers
528 views

Definite integrals with interesting results [closed]

I just stumbled across the fact that $\int_{-\infty}^{+\infty}{e^{-x^2}dx}=\sqrt{\pi}$. This intrigued my already-existing interest in integrals. It made me wonder, are there other integrals with ...
4
votes
0answers
227 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasilinear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
2
votes
1answer
76 views

Convergence of $\int_0^\infty \frac{\log x}{1+x^p}\ dx$

For which values of $p$ does the improper integral $$\int_0^\infty \frac{\log x}{1+x^p}\ dx$$ converge? I tried integration parts and various tricks, but it does not seems to work. Thanks
3
votes
1answer
69 views

Differential equation with a unknown function $g$ but a known solution.

Say we have an equation $y'+g(x)y=3x, x>0$. We know one solution to this FDE: $y(x) = x^2$ How can we know if there's a solution which satisfies $y(1) = 2$? Questions I'd like answered Q1: ...
0
votes
0answers
21 views

How to compute $\int_C (X|dx)$?

I have the vector field $X(x,y,z) = (\frac {-y}{x^2+y^2+z^2}, \frac{x}{x^2+y^2+z^2},0)$ on $\mathbb{R} ^3$ \ $\left\{ (0,0,0) \right\}$. I need to compute $\int_C (X|dx)$ over a closed circle of ...
3
votes
4answers
336 views

Evaluate $\int\sqrt[5]{\frac{x+5}{x-5}}\,\mathrm dx$

How to compute this integral? I stuck at a point where I get $\displaystyle\int\frac{1}{t^5-1}+ \cdots $ $$\int\sqrt[5]{\frac{x+5}{x-5}}\,\mathrm dx$$ using $\displaystyle t=\sqrt[5]{\frac{x+5}{x-5}}$ ...
1
vote
1answer
34 views

Finding the integral using divergence theorem without vector field

Use the divergence theorem to evaluate $$\iint_S (x^2 y^2+y^2 z^2+x^2 z^2)\ dS,$$ where $S$ is the entire surface of the sphere of unit radius, centered at the origin. How do I find this? ...
1
vote
1answer
19 views

Procedure in proving inequalities (or bounds) involving minus sign

Usually when we want to bound an expression involving sums, it is easy to proceed by bounding each term separately since we "do not lose" too much using the triangle inequality $|x+y|\leq |x|+|y|$ ...
1
vote
1answer
257 views

Find the area of the region that is enclosed by the cardioid $r=2+2\sin(\theta)$

We just learned polar integration, so I know that's how we're supposed to do it. I have a problem though: I'm getting a negative answer. What I did: Using the graph, which is: I figured out that ...
1
vote
1answer
36 views

Comparison of the change of variable theorem

I would like to compare the change of variable theorem for 1 variable and more. What are the differences, in which case we need stronger assumptions? How do they differ? What is the best way to write ...
0
votes
1answer
26 views

Bound on expectation of exponentially distributed random variables (RVs)

Let $X_A$ and $X_B$ be independent exponentially distributed RVs with means $\mu_A$ and $\mu_B$, respectively. I have been running simulations that suggest $$\mathbb E \left[ \frac{X_B}{A ...
0
votes
1answer
99 views

Find this integral $I=\int\frac{1}{\sin^5{x}+\cos^6{x}}dx$

How find this $$I=\int\dfrac{1}{\sin^5{x}+\cos^6{x}}dx$$ this problem is from china exam.so I think have closed form I find this [wolf] ...
0
votes
2answers
36 views

How to get the follow integration formula by substitution

I have a formula of pdf: $$f_W (w) = \left( \frac{1}{\mu} \right) \int_{u = 0}^1 \left( \frac{w}{u^2} \right) f_m (w / u) d u$$ How to get the follow integration formula by substitution $y=(w/u)$ ...
0
votes
0answers
66 views

Riemann integrals of abstract functions into Banach spaces

If we define the (Riemann) integral of an abstract function, i.e. a function $f:[a,b]\to Y$ where $Y$ is a Banach space, as$$\int_a^b F(t)dt:=\lim_{\max(t_{k+1}-t_k)\to ...
2
votes
0answers
60 views

Value of improper Integral

I need help in finding the value of the integral $$\displaystyle \int_0^\infty \left(\frac{x^2}{1+x}\right)^{n-1}e^{-tx}dx,$$ where $n$ is a positive integer and $t$ is a positive real number.
2
votes
1answer
44 views

Evaluating $\displaystyle\lim_{n \to +\infty}\int _{-1}^1 f(t)\cos^2(nt) \mathrm dt$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous. Assume that $\displaystyle\int_{-1}^{1}f(t)\mathrm dt=1$. Evaluate $$\lim_{n \to +\infty}\int _{-1}^1 f(t)\cos^2(nt) \mathrm dt$$ How to ...
2
votes
1answer
59 views

Differential equation $y' = -2x y+ e^{-x^2}\sin(x)$

Can someone help me solve this equation? $$y' = -2xy + e^{-x^2}\sin(x)$$ Progress I have been trying to solve it with the method of integrating factor. I get to the point where: ...
0
votes
1answer
84 views

Two definitions of the Riemann-Stieltjes integral

Let us define a partition $a=x_0<x_1<...<x_n=b$ of interval $[a,b]$ and let us define the Riemann-Stieltjes integral $\int_a^b fd\Phi$ of a bounded function $f:[a,b]\to\mathbb{C}$, or ...
5
votes
3answers
70 views

Example of a Countable set which has volume zero.

The following question was asked in my exam which states: Give an example of a countable set which has volume zero. Hint: show that if $S=\{a_1,a_2,\ldots,a_n,\ldots\}$ is the set of points ...
2
votes
2answers
77 views

Evaluate $\int_{-2}^2 (2x^2-4)^4dx$

Evaluate $\int_{-2}^2 (2x^2-4)^4dx$ So, I was trying to solve, I tried to expand this and got something, 347?. Is there like any other way to find the definite integral of this? Because after I ...
0
votes
0answers
27 views

To evaluate a line integral

I need to evaluate line integral where C is a part for which $z\gt0$ and $z=0$ of intersection of surfaces $(x-1)^{2} + y^{2} = 1$ and $x^{2} + y^{2} + z^{2} = 4x$ . I am not writing about vector ...
0
votes
0answers
58 views

Remainimg volume in sphere after cylindrical hole is drilled out from it

Calculate volume remaining after a cylinder of diameter $a$ pierces a sphere of radius $a$ with offset $c$. We may find interesting results in special cases. I know when $c = 0$ (central hole in ...
6
votes
3answers
131 views

Proving $\lim_{n\to \infty}\int_0^\pi\frac{\sin\left(nx\right)}{1+x^2}dx=0 $

$$\lim_{n\to \infty}\int_0^\pi\frac{\sin\left(nx\right)}{1+x^2}dx=0 $$ I consider if it can be solved by other methods without Riemann lemma. I try my best to do it as follow: Let $t=nx$ ...
4
votes
0answers
113 views

Integrate $\int\tan (x)\sin (ax)dx$

$$\int\tan (x)\sin (ax)dx$$ If $a$ integer number I used the wolfram.Alpha.com site to give me the following How can I use this result if I need to compute some values by using a program in ...
3
votes
3answers
103 views

Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$

How to find the Cauchy principal value of the following integral $$\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$$ How to start this problem?