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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2
votes
1answer
66 views

Determine how large the number a has to be?

This is what i've done so far: i converted to limit notation lim as t goes to infinity of integral from a to t of 1/t^2+1 dt lim as to goes to infinity [arctan(t)] from a to t (lim as t goes to ...
0
votes
0answers
22 views

question about simplifying the integral(appeared in the proof of interpolation theorem of BMO and L^p)

I do not know how to calculate the following integral, which I saw when reading a short note about interpolation result about BMO spaces. $r\displaystyle\int_{2^{n/p}}^\infty \lambda^{r-1} c_1 ...
3
votes
1answer
80 views

$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$

While solving $y'=x^2-e^y$ I'm stuck on the last step that requires to evaluate this integral. $$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$$ I don't know how to approach it. I know that it ...
0
votes
1answer
41 views

Integral of $\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$.

Can some one help me with the integral $$\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$$ According to my exercise I should be able to get $0$. Please help me .
11
votes
3answers
408 views

Closed form of $\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$

I have homework to evaluate this integral $$I=\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$$ Here is what I have done so far. I tried integration by parts using $u=\tanh(x)\,\tanh(2x)$ and ...
0
votes
1answer
40 views

search for a theorem related to $\sum\limits_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu$

I need to use the following inequality: $$\sum_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu\tag{1}$$ what is the name of such a theorem?
0
votes
1answer
63 views

What does it mean for an integral to “vanish”?

I had a question; What does it mean for an integral to vanish in complex analysis? There is supposedly something, which says if the integral "vanishes," the sum of the residues is 0. But what does ...
2
votes
1answer
51 views

Integral inequality with sines

I am trying to show that there exists some constant $\mathcal{C}>0$ such that: $$\mathcal{C}\leq \int_0^1 |\sin (2\pi n x)-\sin (2\pi m x)|\;dx$$ For all distinct $m,n\in\mathbb{N}$. The constant ...
8
votes
1answer
161 views

Alternative ways to evaluate $\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$

In the following link here I found the integral & the evaluation of $$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$ I'll also include a simpler version together with the question: is ...
2
votes
1answer
121 views

Solving ODE $F(t)=A(t)F'(t) $

How to solve $F(t)=A(t)F'(t) ,F(0)= I\tag 1$ All are $3 \times 3$ matrices except variable t A(t) is given and has determinant $0$. $A(t)=(I-tC_1)^{-1}t^3C_2 \tag 2$ I is a constant unit ...
1
vote
1answer
25 views

Change of variable integral

Consider the following integral \begin{equation} \int_{\Omega} f(y,My-z_2)\, g(z_1,z_2,y) ~ dz_1\, dz_2\, dy \end{equation} $f(y,My-z_2) = 1$ (a constant function for each value of $y$ and $z_2$) ...
1
vote
6answers
88 views

Solving $\int \frac{1}{\sqrt{x^2 - c}} dx$

I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$ How do I do this? It looks like it is close to being an $\operatorname{arcsin}$? I would have thought I could ...
2
votes
1answer
59 views

Find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$.

Suppose a $X$ is a random variable, I am asked to find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$. I am trying to use the inversion formula for the ...
0
votes
2answers
38 views

How to do this Integration (in Orthonormal Family for Continuous Functions)

Here is a common example in the discussion on orthonormal family. Let $\mathcal L = C[a, b]$. For $k \in \mathbb Z$, let $e_k \in \mathcal L$ be defined by $$e_k(\xi) := \frac{1}{\sqrt{b-a}} ...
0
votes
1answer
62 views

An integral involves Gamma function

Thanks for your attention, I meet an integral involves Gamma function and exponential function as follows:$$\int_a^\infty {{x^\alpha }} {e^{cx}}\Gamma \left( {s,bx} \right)dx$$ where $a > 0,s ...
1
vote
3answers
78 views

Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge?

The Question Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge? Please note I have no knowledge of Alternating Series, Ratio and Root tests, Power Series, or Taylor and McLaurin Series. My Work ...
1
vote
1answer
38 views

Curve parameterization trick

So, I was given this really nasty problem to solve Suppose C is parametrized by $\mathbf{g}(t) = \left[\begin{array}{c}e^{t^{3}\cos\!\left(2\pi t^{25}\right)}\cr t^{6}+3t^{3}+3\cr ...
3
votes
2answers
111 views

What is the correct definition of Area?

How is the area of a rectangle: length $\times$ breadth? We know that other areas can be derived from it. Also, the area under curves uses the area of rectangles as a basis.
1
vote
0answers
95 views

Triple integral-pyramid

Let the pyramid with vertices $A(0,0,0), B(0,0,1), C(0,1,0), D(1,1,0)$. I need to find the equations of the four planes bounding the pyramid then I have to set up an integral for the volume in three ...
6
votes
3answers
118 views

How to evaluate this improper integral?

I got stuck when evaluating these two improper integrals:$$ \int_a^b\frac{dx}{\sqrt{(b-x)(x-a)}} $$ and$$ \int_0^1\frac{dx}{\sqrt{x-x^3}} $$ How to evaluate them? Thank you!
1
vote
2answers
59 views

Mean of a Cauchy Distribution

Why is the mean of a Cauchy distribution undefined? Surely, it should be $0$ by symmetry? $$\int_{-\infty}^{\infty} {\frac{x}{\pi (1+x^2)}} dx =0?$$
1
vote
2answers
45 views

Is $f\left(t\right)=\frac{1}{t^2+1}$ of exponential order?

I'm learning Laplace Transforms and one of the questions I'm working on is the following: $$\text{Is}\:\:f\left(t\right)=\frac{1}{t^2+1}\:\:\:\text{of exponential order?}$$ If so or if not, how do I ...
2
votes
3answers
99 views

limit $\lim_{n\rightarrow \infty}\left(\int_0^1f^n(x)dx\right)^{\frac{1}{n}}=M$

For a continuous positive funciton $f(x)$ on $[0,1]$, with maximum value $M$, show that $$\lim_{n\rightarrow \infty}\left(\int_0^1f^n(x)dx\right)^{\frac{1}{n}}=M$$
0
votes
1answer
51 views

Shifted Fourier transform

Please can some one help and give me a direction to evaluate the following shifted Fourier transform: \begin{alignat}{2} s(x_c) =&\frac{1}{\Delta x_0} \int_{x_c-\Delta x_0}^{x_c+\Delta ...
0
votes
3answers
30 views

integration help with area

Find the area of the region bounded by the curves of $y=2x^2-3x+5$ and $y=x+11$. So far I have done this: $2x^2-3x+5 = x+11$ $2x^2-4x-6=0$ $2(x^2-2x-3)=0$ $(x-3)(x+1)=0$ $X=3, X=-1$ What do I ...
2
votes
2answers
47 views

Integral $\int_{\pi/2+\delta}^{3\pi/2-\delta} x^{R \cos \varphi} d \varphi$ bounded

This is probably a silly question, or maybe I am missing a very simple slick trick, but I am trying to see how the following integral is bounded in terms of $\delta$: \begin{equation} ...
-1
votes
1answer
34 views

Reversing the integration order

I believe the answer would be like this: $$ \int_{-\sqrt{2-y^2}}^{\sqrt{2-y^2}}\int_{1}^{\sqrt{2}} f(x,y) \, dx \, dy. $$ If anyone could check my work and point me in the right direction that ...
3
votes
3answers
756 views

Wolfram Alpha can't solve this integral analytically

Wolfram Alpha isn't able to calculate this integral (I don't have mathematica, but I have Wolfram Pro). $$\int_{0}^{a} \frac{1}{\sqrt{(x-a)^2+(x-b)^2}} \ dx \ \ \ , \ b>a$$ This is for a physics ...
0
votes
1answer
69 views

I'm having a tough time with this integral (Magnetic Vector Potential)

Wolfram Alpha isn't able to calculate this integral (I don't have mathematica, but I have Wolfram Pro). $\int_{0}^{a} 1/\sqrt{(x-a)^2+(x-b)^2}dx$ $;b>a$ This problem comes from solving for the ...
7
votes
0answers
150 views

Evaluating $\int_0^{\infty} \log(\sin^2(x))\left(1-x\operatorname{arccot}(x)\right) \ dx$

One of the ways to compute the integral $$\int_0^{\infty} \log(\sin^2(x))\left(1-x\operatorname{arccot}(x)\right) \ ...
36
votes
2answers
804 views

A strange integral

While browsing on Integral and Series, I found a strange integral posted by @sos440. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because ...
2
votes
0answers
27 views

Riemann Integrals and Periodic Functions

Consider the following result, where integrals are, say, Riemann integrals: Let $f:\mathbb{R}\to\mathbb{C}$ be a periodic function of period $L$. Then for all $a\in\mathbb{R}$ $$ ...
0
votes
0answers
44 views

Help to find the solution of an integral

Is there a way to compute the following integral? $$\int_{x \in \mathbf{R}} \exp\left(-\frac{x^2}{2} \right) \int_{-\infty}^{c_1+c_2 x+c_3 x^3} \exp\left(-\frac{t^2}{2} \right) ...
1
vote
1answer
102 views

calculating the taylor term of an integral

an exercise ask me to calculate the Taylor term at $x = 0$ and degree four. I know how to take a derivative of an integral, but I'm having doubts about this one. The function: $$\int_0^x e^{-t^2} ...
0
votes
2answers
26 views

Finding the volume when the region between $y=x^2$ and $y=4-x^2$ is rotated about the $x$-axis.

So I want to find the volume when the region between $y=x^2$ and $y=4-x^2$ is rotated about the $x$-axis. So I start by finding the roots where they meet, so I find : $$\int_{ -\sqrt{2} ...
7
votes
1answer
65 views

Computing $ \int \frac{{x}~{\cos^{-1}(x)}}{\sqrt{1-{x^2}}}~\mathrm{d}x $.

I've just begun to learn integration which makes me a little nervous! Here's a question I'm having a problem with. Also my first time trying to use LaTeX. I apologise for any discrepancies. ...
1
vote
1answer
34 views

The concept of integrating along a square?

I had a question; What is the idea (in complex analysis) of integrating along a square? Take a look at @M.N.C.E.'s method on Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$ I am not quite sure what ...
3
votes
1answer
44 views

Why does the whole integral converge but not part of it? (Dilogs)

$\newcommand{\Li}{\operatorname{Li}}$Consider the integral: $$\int_0^1 \frac{(-\Li_2(x) - \Li_3(x) - x^2/8 + 3x - x\log(1-x) + \log(1-x))}{x^2} \, dx$$ This integral converges to $\sim 0.01$ But ...
0
votes
2answers
70 views

Evaluate ${M\over2R^2}\int(R^2 - y^2)^{3\over2}dy$

I was finding the moment of inertia of a hollow sphere, and got stuck on the integration of: $${M\over2R^2}\int(R^2 - y^2)^{3\over2}dy$$ Any hint to as to solve this integral? Please help. ...
0
votes
1answer
68 views

Integrability of derivative of indefinite integral

Let $G(x)=\int_a^xg(t)dt$ for some $g(t):[a,b]\rightarrow \mathbb{R}$ ($g(t)$ is Riemann integrable on $[a,b]$) , and suppose we know that $G(x)$ is differentiable on $[a,b]$. Does it follow that ...
0
votes
1answer
60 views

Challenging Question: for Expected Value of a particular probability density function

I've been stuck on this for a while and it's been driving me crazy. Any help would be greatly appreciated. I am trying to find the Expected Value of the following Probability Density Functions (where ...
2
votes
0answers
95 views

2 logarithmic integrals twins

This question also has the value of an answer to the first integral here How to evaluate $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1-x}dx$ and $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1+x}dx$ ...
0
votes
1answer
18 views

Density of a plate, u-sub?

I get the problem and can set it up but am struggling with the integration. Is there u-substitution and I just can't figure it out?? THE PROBLEM A thin metal plate occupies a region D, which lies in ...
1
vote
1answer
35 views

Triple Integral in Cartesian, Cylindrical and Spherical

We have a conical solid bounded by the surface: $z=2 \sqrt{x^{2}+y^{2}}$ and $z=2$ where $R=1$ and $H=2$ set up the integral in: 1) Cartesian (in the order $dzdydx$) 2) Cylindrical (in the order ...
0
votes
2answers
41 views

function with equal upper and lower sums

Which function f on [0,1] has equal lower and upper sum for each partition? I guess constant functions are the only such functions. Give me some hint to prove it. Thanks in advance.
0
votes
0answers
93 views

Calculating the lower and upper Darboux Sums for a given partition

So consider the partition $P=\{0,1/4, 1/2, 1\}$ and function $f(x)=-x^2$, where $x\in[0,1]$. How to find the $L(f,P)$ and $U(f,P)$, namely the lower and upper Darboux's sums. My calculation is as ...
1
vote
1answer
64 views

Prove $\int_0^\pi\sin^{2n}t dt$ without using Residue Theorem

How may one prove something similar as in here but from $0$ to $\pi$ and without using the Residue Theorem? I was told to consider the contour integral $$\int_{|z|=1}(z-\frac{1}{z})^{2n}dz/z $$ and ...
0
votes
1answer
69 views

Matrix exponential - From given Infinite series result

Given Data in the question We have a recursion and its sum defined as follows $ \left\{ \begin{array}{ll} R_0(t)=I_{3 \times 3} & \mbox{if } n = 0 \\ R_1(t)=tR_0 \hspace{.1cm} A ...
2
votes
0answers
39 views

Computation of double integral $(1-xy)^b$

I want to determine the values of $b>0$ such that $$\int_0^1\int_0^1(1-xy)^{-b}dydx$$ exists and is finite. I think that the integral is finite for $0<b<2$ and infinite for $b\geq 2$ but I am ...
3
votes
3answers
156 views

How to calculate $\int_0^\infty \frac{dx}{1+x^6}$ [duplicate]

Whenever I tried to do, it failed. Is there anyone to help? $$\int_0^\infty \frac{dx}{1+x^6}$$