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Questions tagged [integration]

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 ...

2
votes
2answers
28 views

Compute complex integral inside an open curve

I need to compute this complex integral: $$ \int_\gamma \frac{1}{(z-i)(z-2i)} dz $$ $\gamma$ is defined as: $$ \gamma (t) = t + i(3e^t\cos^2(t)) $$ The parameter t belongs to the following ...
2
votes
1answer
25 views

Integration of probabilities - decision theory - minimizing misclassification rate

I have trouble understanding an equation in a book I'm reading. Basically, Consider a decision rule that divides the input space into regions $R_k$ called decision regions, one for each class, such ...
-2
votes
0answers
31 views

Double integration: $\sin(2y+3x)$. [on hold]

I have to solve this double integral: $$\int_C \sin(2y+3x)\,dx\,dy$$ where $C$ is the sector of the circle with radius $1$ centered at the origin between the angles $\pi/2$ and $3\pi/4$.
1
vote
1answer
18 views

Simulation: Generate random numbers that cluster around an average?

I want to simulate a simple event that has variable empirical result/outcome. Generate random numbers that cluster around an average For example, let's say we collect the data for how far people can ...
0
votes
0answers
10 views

Drawing a saddle on a contour graph

When drawing a contour graph of say, $f(x,y) = \frac12x^2 + \frac13y^3 - xy - x + y$ , there is a saddle point at point $(1,0)$. How can I determine the slopes of the two lines that meet at this ...
-1
votes
2answers
36 views

Indefinite integration by parts

I was given an excersice to solve wich asks you to prove: $$\int_0^1 f(r) r dr = 0 $$ knowing that: $$\int_0^1 f(t) dt = 0 $$ After doing integration by parts I ended up with: $$\int f(r) r dr = ...
0
votes
0answers
14 views

Suggestions for solving a complex triple integral

I am trying to solve the following integral: $\frac{1}{\sigma_1}\int_{-\infty}^{\infty}\phi\Big(\frac{y_{11}-\mu_{11}}{\sigma_1}\Big)\Big(BvN(h_{11},(y_{11}+\tau_1);\rho_3)-BvN(h_{12},(y_{11}+\tau_1);...
1
vote
1answer
27 views

How to show the function $f(x) = x^2 \sin(1/x)$ has integrable derivative?

Consider the function $$f(x) = \begin{cases} x^2\sin(1/x) & \text { if } x \neq 0 \\ 0 & \text{ otherwise} \end{cases} $$ has integrable derivative on $(-1, 1)$. I found $$f'(x) = \begin{...
4
votes
1answer
26 views

On Harmonic numbers at half-integer values

Harmonic numbers are usually defined, for $n\in\Bbb N$, by $$H_n=\sum_{k=1}^{n}\frac1k$$ But then one may note, $$H_n=\sum_{k=1}^{n}\int_0^1x^{k-1}\mathrm dx=\int_0^1\frac{1-x^n}{1-x}\mathrm dx=\int_0^...
-1
votes
1answer
31 views

What is the integral of $\frac{\mathrm{d}^2f(x)}{\mathrm{d}x^2}$ [on hold]

Could you please explain how can I integrate the following $$\frac{\mathrm{d}^2f(x)}{\mathrm{d}x^2}$$ where $y\equiv f(x)$.
3
votes
4answers
48 views

Prove $\arcsin(1)<\int_0^b1/\sqrt{1-x^2}dx +(1-b)\pi/2$

I'm trying to prove the following inequality: $$\arcsin(1)<\int_0^b1/\sqrt{1-x^2}\,dx +(1-b)\pi/2$$ for every $b \in [0,1)$. I'm given $\arcsin(1) = \pi/2$ and $\arcsin(x)$ is strictly ...
-4
votes
1answer
34 views

I need help with this integral $\int_{-1}^{1} \lfloor 4 x^{2}\rfloor dx$ [on hold]

I don't know how to evaluate the parts of this function, the rest is a piece of cake $$\int_{-1}^{1} \lfloor 4 x^{2} \rfloor dx$$
0
votes
2answers
24 views

Integration of $\int_0^\frac{\pi}{6} \cos^{-3}2x \sin2x \,\ dx $?

I tried substituting $x=\frac{\cos t}{2}$ but I didn't got anywhere. Thanks!
0
votes
2answers
33 views

Integral of 1/x - base of logarithm

I see a proof in https://arxiv.org/abs/1805.11965 (equation 3.36) that uses the following. $\log x = \int_0^{\infty} ds \left(\frac{1}{1+s} - \frac{1}{s+x}\right)$. This seems to hinge on $\int \...
4
votes
1answer
43 views

Find function $f(x)$ that satisfying differential relation

Suppose the functions $F(x)$ and $G(x)$ satisfying $$F(x)=f(x)-\frac{1}{f(x)}$$ $$G(x)=f(x)+\frac{1}{f(x)}$$ such that $F'(x)=(G\circ G)(x)$, with initial condition $f(\frac{\pi}{4})=1$ is given....
0
votes
0answers
14 views

Satellite radiance measurements in spectral regions

Q: A satellite measures longwave radiation emerging from a planet with uniform surface temperature T1 and uniform atmospheric temperature T2. What values of radiance does the satellite measure in the ...
1
vote
1answer
18 views

Almost volume preserving charts for a riemannian manifold

Let $M$ be riemannian manifold. Is it true that for all $p\in M$ and $\varepsilon>0$ there is a chart $\phi:U\rightarrow\mathbb{R^n}$ arround $p$ such that for all nonempty open $V\subseteq U$ $\...
0
votes
0answers
39 views

How to solve a Convolution Integral with one delta function.

I have a Convolution integral $$ \int_{t_0}^{t} \int_{t_0}^{\tau} C(t-t')C(\tau -t'') \delta(t''-t') dt'' dt'=\int_{t_0}^{t} C(t-t')C(\tau -t') dt'= ? $$ I do not know how to proceed any further, $\...
1
vote
1answer
49 views

How to calculate the following integrals?

How the calculate the following integrals? Therein $D$ is a constant. $$(1)\;\;\int_{0}^{2\pi}\frac{1}{1-D\cdot\cos\theta} d\theta$$ and $$(2)\;\;\int_{0}^{2\pi}\frac{1}{1+D\cdot\cos\theta} d\theta$...
0
votes
0answers
8 views

Cross Product in 3D Cylindrical or Spherical coordinates

When i take a cross product of two vectors (Cylinder base in x,y plane) for example d(phi)cross d(z), how do i know if the resultant Vector protrudes out of the page or goes inside? I know the right ...
1
vote
0answers
18 views

Darboux sums elementary question - am I correct

I'm very new to this material and I would like someone more experienced to give input if possible. $Q \subset \mathbb R^n$ is a box and $f: Q \to \mathbb R$ is a function. Let $\Xi_Q$ be the set of ...
0
votes
0answers
15 views

$g_t = \frac{e^{f_t}}{\int e^{f_t}}$ is differentiable

Let $\Omega$ be a metric space and $f_0, f_1: \Omega \rightarrow \mathbb{R}$ be any two differentiable functions. For each $t \in [0,1]$, define the function $f_t = tf_1 + (1-t)f_0$. I need to show ...
2
votes
1answer
55 views

Reimann zeta function

I started solving a problem using the transformation of Reimann zeta function into this form: And I searched for the methods of doing this transformation and ended up with this one : It is a little ...
-5
votes
0answers
32 views

Compute an integral over $\mathbb C$ [on hold]

By using polar coordinate $z=re^{i\theta}$, a simple computation give $$\int_{\mathbb C}|z|^{2n}e^{-|z|^2}dz=\Gamma(n+1)=n!.$$ Now, I would like to compute the following integral: $$\int_{\mathbb C}|...
1
vote
1answer
16 views

Find the marginal distribution of an point randomly chosen on an ellipse

This exercise comes from rice 3.6 and states: A point is chosen randomly in the interior of an ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Find the marginal densities of the $x$ and $y$ ...
1
vote
0answers
31 views

Calculate the expectation of random variable, integration of a function.

I want to calculate the expectation of one random variable $\frac{1}{\sqrt{a^2+x^2}}$, where $x\sim N(0,\sigma^2)$ and $a$ is a constant. It is straightforward that we can come to the integration of ...
0
votes
0answers
19 views

Transfer Gauss 2 point rule [-1,1] to [0,1] [on hold]

Given that Gauss 2 point rule on [-1,1] is the following: Where w0 = 1, w1 = 1, x0 = −(√3)/3, and x1 = (√3)/3 How can I transfer the rule from interval of [-1,1] to [0,1]?
0
votes
0answers
17 views

Determining constants to make a quadrature formula with degree of precision equal to three.

I just need an explanation on how these linear equations are found when plugging and testing $f(x) = x^k. $ For example, in the first linear equation, what happens to the constants c and d. Why are ...
2
votes
1answer
20 views

Calculate this triple integral in cylindrical coordinates, the result is different with triple integral in cartesian coordinates

I want to calculate triple integral \begin{equation}\int\limits_{-1}^{1}\int\limits_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int\limits_{x^2+y^2}^1 2z dzdydx.\end{equation} (the surface is $z=x^2+y^2$, $0\leq z\...
0
votes
0answers
20 views

How to evaluate this surface integral?

I am given the following problem: Use a change of variables to evaluate $\int \int_T e^{(y-x)/(y+x)} dA$ where $T$ is the triangular region with vertices $(0,0)$, $(1,0)$ and $(0,1)$. Here is what ...
1
vote
3answers
32 views

volume of surface given by $(x^2+y^2+z^2)^2=x$

A question asks me to find the volume of the surface $(x^2+y^2+z^2)^2=x$ this looks like a very difficult triple integral to evaluate using cartesian coordinates, so I though I would describe the set ...
0
votes
0answers
27 views

How to find pdf of X+Y given X and Y are dependent.

The joint pdf is f(x,y) = $$\frac{2}{5}(2x+3y)$$ for $0\leq x \leq 1,0\leq y \leq 1$ Normally if the random variables are independent, you can apply the convolution definition Z = X + Y which looks ...
-1
votes
1answer
32 views

Calculate the double integral.

First of all I would like to ask you if you know a very good material that could help me with range of integration. $$\int_{-1}^ {1} \int_{0}^{x+2}y\,dy\,dx$$ How do I solve this question?
0
votes
1answer
42 views

How to get derivative of integral with $\ln(x)$?

Find $f'(x)$ if $f(x) = \int_1^{ln(x)} e^{t^2} \,dt$ The correct way to solve it: $$f'(x) = e^{(\ln{x})^2} \frac{1}{x}$$ $$f'(x) = \frac{1}{x}e^{(\ln{x})^2}$$ I haven't seen an example like this ...
3
votes
1answer
51 views

recurrence relation for $\zeta(2n)$

I found this formula. Is it correct? For $n\in\Bbb N,\ n\geq2$, $$\zeta(2n)=\frac{2n\pi^{2n}}{\Gamma(2n+2)}+\sum_{k=0}^{n-2}(-1)^{k-n}\frac{\pi^{2n-2k-2}}{\Gamma(2n-2k)}\zeta(2k+2)$$ Here's my proof....
0
votes
1answer
38 views

Solving integral with absolute value

With a given $x > 0$ (I think we could restrict it to $x \in [0, 3]$), I'm trying to find the following integral: $$ \int_{z = 0}^{\min(x,1)} \int_{y = 0}^{\min(x - z, 1)} |x -z -y -1| dydz $$ ...
3
votes
0answers
39 views

Show: $\lim_{n\rightarrow \infty} \left|\int_{1}^{e}\left[\ln(x)\right]^n\:dx \right|= 0 $

This is probably an unremarkable result, but I was curious about exploring the following family of definite integrals to see what asymptotic behaviour it as as $n \rightarrow \infty$ $$I_n = \int_{1}^...
0
votes
0answers
18 views

Solution $y(x)$ for $-\cos(x)= \int_0^{2\pi}\max(y(t), y(x+t))dt$

The application here is to design a value which will produce a sine wave like pressure or flow rate as a function of time. Pressure, or flow rate is a function of the open area inside of a valve. ...
2
votes
2answers
21 views

Comparing integral and series of $1/(x^a)$

The problem is to $$\sum^N_{n=2}\frac{1}{n^a}\leq\int^N_1\frac{1}{x^a}$$ and to use this to prove the convergence of the series for $a>1$. So, I believe I have the second part down. Namely ...
3
votes
3answers
70 views

The integral $\int_{-\infty}^{\infty}\frac{1}{R^2+z^2}e^{-\alpha \sqrt{R^2+z^2}}\mathrm{d}z$

I need to evaluate the following integral (for a physical application): $$I=\int_{-\infty}^{\infty}\frac{1}{R^2+z^2}e^{-\alpha \sqrt{R^2+z^2}}\mathrm{d}z, $$ where $\alpha>0$ and $R>0$. I tried ...
-1
votes
0answers
38 views

Why is $d/du(\int_{1}^{u}{\sec(t)}\,dt) = \sec(u)$? [on hold]

$$d/du(\int_{1}^{u}{\sec(t)}\,dt) = \sec(u)$$ This is a step in finding the derivative of the more complicated integral: $$ f(x) =d/dx(\int_{1}^{x^4}{\sec(t)}\,dt)$$ As you can see the step involves ...
3
votes
2answers
54 views

Evaluate $\int_{-\infty}^\infty \frac{1}{\sqrt{z^2 + 1}}\frac{1}{z - \alpha} dz$.

Evaluate $$\int_{-\infty}^\infty \frac{1}{\sqrt{z^2 + 1}}\frac{1}{z - \alpha} dz\,.$$ What is an elegant way to evaluate this integral for Im $\alpha >0$? I imagine using residue theorem will lead ...
2
votes
1answer
59 views

How to calculate $\int_0^{\pi/2} \sin^a x \cos^b x \,\mathrm{d} x$

$$\int_0^{\pi/2} \sin^a x \cos^b x \,\mathrm{d} x = \frac{\Gamma\left(\frac{1+a}{2}\right)\Gamma\left(\frac{1+b}{2}\right)}{2 \Gamma\left(1 + \frac{a + b}{2}\right)} \quad\quad\text{for } a, b > -1$...
-1
votes
3answers
71 views

Help to calculate the integral $\int 314^{\cos x} \; dx$

I stopped at the place highlighted in yellow how to find this integral ? $$\int (314)^{\large \cos x} \; dx$$ $$\begin{aligned}\int 314^{\cos x}\sin x\,\mathrm{d}x=&\int u\mathrm{d}v=uv-\int v\...
-2
votes
1answer
45 views

Does $ \int_0^{+\infty} \frac{e^{-x}}{x^{\alpha}}dx$ converge or diverge?

Does this integral converge or diverge? $$ \int_0^{+\infty} \frac{e^{-x}}{x^{\alpha}}dx\qquad (\alpha \geq 0) $$ I was stuck when solved this problem this way: $ \int_0^{+\infty} \frac{e^{-x}}{x^{\...
0
votes
1answer
40 views

When do triple integrals involve a fourth dimension?

Of course, heuristically, a single integral gives area under a curve, and a double integral of a function gives the volume under the integrand and above a two-dimensional domain. Now, I understand ...
-4
votes
0answers
24 views

Solve the following integration and find the answer [on hold]

$ \frac{1}{\sqrt{2\pi}}$ integral from $0$ to $\infty$ $\cos tx e^{\frac{−x^{2}}{2}} dx$ Help me to solve this integration... The answer is $e^{\frac{-t^{2}}{2}}$ I want the steps
2
votes
0answers
24 views

Gaussian-like integration by parts on the high-dimensional hypersphere

Denote $\mathbb{S}_n(\sqrt{n})$ the sphere in $\mathbb{R}^n$ with radius $\sqrt{n}$. Recall Stein's lemma (Gaussian integration by parts), which reads for every function $f_n$ such that the ...
0
votes
0answers
10 views

To find the moment of inertia about co-ordinate axes of a tetrahedron in the first octant bounded by x+y+z=1

I need to find the moment of inertia about co-ordinate axes of a tetrahedron in the first octant bounded by $x+y+z=1$. My attempt: $x+y+z = 1$ forms the base of the tetrahedron and my limits for ...
-2
votes
0answers
35 views

Is this function Lebesque-Integrable? [on hold]

Let $\alpha$ be a real integer. We define the mapping $f : \mathbb{R}^{n} \setminus \{0\} \rightarrow \mathbb{R}$ where $f(x) = \alpha \|x\|.$ Show for which $\alpha$ is $f$ Lebesgue-integrable on $...