Tagged Questions
Questions about the evaluation of specific definite integrals.
0
votes
2answers
36 views
Having trouble with differentiating under integral sign
I am sorry if this seems like a dumb question, but I am having trouble in applying differentiation under the integral sign to definite integrals such as this one:
$$\int^{1}_{0} ...
2
votes
6answers
113 views
What is an easy way to integrate $\int_0^5 \frac{v^3 }{2\sqrt{25-v^2}} dv$?
This does not appear to be a difficult integral.
I am wondering if there was an easy way to do it.
0
votes
0answers
53 views
Did I really fail the Gram Shmidt procedure, or the website had an error? linear algebra
below is the results from my website assignement. There is also my handwritten work-up to my answers. I don't see any error Ii made in finding "C", could the website have an error?
8
votes
2answers
110 views
Finding the maximum value of $\displaystyle \int_{0}^{1}e^x\log f(x)dx$ when $\displaystyle \int_{0}^{1}f(x)dx=1$
Suppose that $f(x)\ (0\le x\le 1)$ is continuous and strictly positive and satisfies $$\int_{0}^{1}f(x)dx=1.$$
Then, can we find the maximum and the minimum value of the following? If yes, then how?
...
2
votes
4answers
447 views
Integration by substitution, why do we change the limits?
I've highlighted the part I don't understand in red. Why do we change the limits of integration here? What difference does it make?
Source of Quotation: Calculus: Early Transcendentals, 7th Edition, ...
3
votes
1answer
48 views
Difficult infinite integral involving a Gaussian, Bessel function and complex singularities
I've come across the following integral in my work on flux noise in SQUIDs. $$\intop_{0}^{\infty}dk\, e^{-ak^{2}}J_{0}\left(bk\right)\frac{k^{3}}{c^{2}+k^{4}}
$$
Where $a$,$b$,$c$ are all positive.
...
2
votes
1answer
87 views
Use discrete proof to show that $\int f^2 \int g^2 \geq (\int fg)^2$
One proof of the Schwarz inequality on $\mathbb{R}^n$ is to note that $$(\sum x_i^2)(\sum y_i^2) = (\sum x_i y_i)^2 +
\sum_{i<j}(x_iy_j - x_jy_i)^2.$$ Spivak's Calculus, 4th ed., exercise ...
4
votes
3answers
622 views
The area of circle
The question is to prove that area of a circle with radius $r$ is $\pi r^2$ using integral. I tried to write $$A=\int\limits_{-r}^{r}2\sqrt{r^2-x^2}\ dx$$
but I don't know what to do next.
1
vote
2answers
121 views
Double integral of a rational function
Consider the region $D$ given by $1\leq x^2+y^2\leq2\land0\leq y\leq x$. Compute $$\iint_D\frac{xy(x-y)}{x^3+y^3}dxdy$$
Attempt: The region $D$ is part of a ring in the first quadrant below the line ...
3
votes
3answers
58 views
How do i prove $\int_{0}^{2a}f(x) dx = \int_{0}^{a}f(x)dx +\int_{0}^{a}f(2a-x)dx$
I am stuck on this one, I was able to prove $\int_{0}^{a}f(x) dx = \int_{0}^{a}f(a-x)dx $
just by using substitution with $t = a - x$, but I'm not sure about this one, I've tried quite a few ...
8
votes
3answers
140 views
Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
I have in trouble for evaluating following integral
$$\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$$
It seems really easy, but I ...
2
votes
3answers
145 views
Trouble computing this double integral
$$\iint_R xe^{xy}~\mathrm{d}A \qquad 0\le x\le 2 \quad 0 \le y \le 1$$
Today I started learning about double integrals on a class I am taking, had good understanding on single-variable integrals but ...
9
votes
2answers
176 views
A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$
We already know that
\begin{align}
\displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3),
\\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
5
votes
1answer
76 views
Evaluate: $I = \int^{\pi/2}_0 (\sqrt{\sin x}+\sqrt{\cos x})^{-4}dx$
Evaluate : $$I = \int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx$$
Attempt :
\begin{align}
I&=\int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx\\
...
3
votes
3answers
108 views
Find $\lambda$ if $\int^{\infty}_0 \frac{\log(1+x^2)}{(1+x^2)}dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}dx$
Problem : If $\displaystyle\int^\infty_0 \frac{\log(1+x^2)}{(1+x^2)}\,dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}\,dx$ then find the value of $\lambda$.
I am not getting any clue how to proceed ...
1
vote
0answers
49 views
Analytical evaluation of integral
I would like to evaluate the following integral analytically, but Mathematica does not give me an answer:
$$
\int_0^1 dr \ e^{(1-2r)x^2} \left[p(r,x) Y_0\left(2x^2\sqrt{r-r^2}\right)+q(r,x) ...
0
votes
1answer
49 views
How do they integrate this exponential?
Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an ...
4
votes
3answers
81 views
Proving $\int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1$.
By testing in maple I found that
$$
\int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1
$$
Does there exists a proof for this? I tried rewriting it as an series but no luck ...
3
votes
0answers
38 views
Prove that $I = \int_0^{m(m+1} y_n(x)\,\mathrm{d}x$ converges and $I \in \mathbb{Q}$.
My problem is stated as follows
Let $y_0(x) = x, \ \: y_1(x) = \sqrt{x}, \ \: y_{n+1}(x) = \sqrt{y_n(x) +x\,} \ $. Now define
$ \displaystyle \hspace{3cm} I_n = \int_0^k ...
1
vote
0answers
28 views
How to establish the equivalence of these two statements about integrals of step functions?
First Statement:
Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
29 views
How to establish this equivalence for integrals of step functions?
First Statement:
Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$
Second Statement:
Let $s$ be ...
0
votes
1answer
27 views
exchanging partial derivative and an integral
Does $\frac{\partial }{\partial x}\int_{u(x)}^{v(x)}f(x,t)dt$ = $\int_{u(x)}^{v(x)}\frac{\partial }{\partial x}f(x,t)dt$ ??
If yes, how did we do that although there are 2 functions $u$ and $v$ ...
7
votes
2answers
159 views
Integral: $\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx$ for $a,b>0$
I tried this:
$$\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx=\Re\left(\int_0^{\infty} e^{-ib^2x^2+ia^2/x^2}\,dx\right)=\Re\left(\int_0^{\infty} ...
6
votes
3answers
93 views
A closed form expression for $\int_0^{\infty} (t^2+t^4)^n e^{-t^2-t^4}\,dt$
I was doing some computations for research purposes, which led me to this integral:
$$I(n) = \int_0^{\infty} (t^2+t^4)^n e^{-t^2-t^4}\,dt.$$
This is very suggestively written so as to employ a ...
0
votes
1answer
22 views
How to identify continuity or discontinuity of an [Definite] integral?
How can I figure out whether an improper integral converges based on the discontinuities in the integrand?
For instance, these two both have discontinuities within the intervals of integration, and ...
0
votes
0answers
15 views
Integrals depending upon a parameter
There was an exercise, in my professor's book, asking to prove the continuity of an integral depending upon a parameter. Namely, the hypothesis were:
Let $D$ be a measurable subset of $\mathbb{R}^n$, ...
0
votes
0answers
15 views
Quadrature methods: even order?
I noticed that all quadrature methods I know (Newton-Cotes and Gaussian quadrature) have always even order in the sense that a quadrature method is of order $n$, if all polynomials of degree $n-1$ are ...
1
vote
1answer
62 views
How to calculate an integral
I wonder how the integral
$$\int_{-1}^1 \! \int_0^{\sqrt{1-x^2}} \! \int_0^{\sqrt{1-y^2-x^2}} \! 1 \, dz \, dy \, dx $$
Any ideas?
3
votes
0answers
42 views
On integration of a Gaussian-like function over a region $g(\mathbf{x})\leq 1$
Let $X$ be a random variable which follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the symmetric positive definite $n\times n$ matrix ...
-2
votes
1answer
60 views
Integral question challenge [duplicate]
I try to find a reasonable solution for this equation but i couldent
I try to study lots of material but i couldent solve it. I am a high school student and try to learn.
Integral cos(log x)dx
-1
votes
3answers
108 views
Integral big question
Anyone could help me to solve this equation
I try to study lots of material but I coulden't solve it. I am a high school student and try to learn.
$\displaystyle\int \cos(\ln(x))dx$?
4
votes
2answers
58 views
2D Integral of Bessel Function and Gaussians
I've run into the following integral, and I'm not sure how to evaluate it.
$$F(k)=\int ...
1
vote
1answer
44 views
Limit of the integral: $\int_0^{\pi/2}\beta^\alpha\exp\left(-\beta\cos(\theta)\right)d\theta$
I have the following integral:
$$\displaystyle J(\alpha,\beta)=\int_0^\dfrac{\pi}{2}\beta^\alpha\exp\left(-\beta\cos(\theta)\right)d\theta$$ where: $\alpha\gt0$, $\alpha\in\mathbb{R}$ and ...
0
votes
0answers
10 views
Order of Romberg's method
We call a method(numerical integration) of $n-$th order, if it can integrate any polynomial of degree $n-1$ without any error.
In this sense: The simpson rule is of $4$-th order and the trapezium ...
4
votes
6answers
104 views
If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges
I know the Initiative answer, can anyone give a neat answer based on solid reasoning
EDIT : $f(x)$ is continuous
0
votes
4answers
131 views
How large should $a$ be so that $\int_a^{\infty} \frac{dx}{1+x^2} < \frac{1}{1000}$
I want to solve this without using calculator.
3
votes
0answers
107 views
${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$
I have found the following new result connecting two rational log-cosine integrals.
Proposition. \begin{align}
\displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
0
votes
1answer
25 views
Normalizing a probability density function
I need to find a normalization term $N(\alpha,\beta)$ for the probability density function:
$$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$
In other words, solve the following equation:
...
2
votes
1answer
78 views
Is there an analytical solution to Gaussian integral $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx$?
I wonder if there is an analytical solution to
$$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx,$$
where $a, b>0$.
I know, of course, that the antiderivative of the fraction is a version ...
2
votes
3answers
92 views
Find $x > 0$ for which $\int_{0}^{x} [t]^2 \ dt = 2 (x-1)$.
What are all possible $x > 0$ for which the following equation is satisfied?
$$\int_{0}^{x} [t]^2 \ dt = 2 (x-1),$$ where $[.]$ denotes the bracket (or floor) function.
I guess we will have to ...
1
vote
1answer
44 views
How to solve this seemingly simple triple integral?
$$\iiint_D x^2+y^2+z^2\,dxdydz$$ $D$ is bound by $x=0, y=0,z=0$ and $x+y+z=a$, calculated by rote, I got $\frac{a^5}{20}$, is there any simpler way to do this? I tried using spherical coordinates, but ...
5
votes
2answers
149 views
Prove that $f$ is constant on $[a,b]$
$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$
And $f$ is continious on $[a,b]$ and ...
1
vote
2answers
22 views
The volume is preserved by the flow: where is the absolute value?
Consider the following excerpt of the Liouville's theorem proof taken from "Arnold - mathematical methods of classical mechanics":
In changing the variables in the integral, I don't understand why ...
0
votes
0answers
48 views
Solution to the integral?
What is the solution of the following integral:
$$
\int_{-1}^{K} x^{B+1} e^{-Nx} dx
$$
where $N$ and $B$ are constants
1
vote
0answers
91 views
+50
$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$
The following integral bothers me since weeks:
$$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$$
Has any body a suggestion for this integral.
$Re\; s >0$ sufficiently large and $s'$ an ...
1
vote
1answer
32 views
consider a square of side length $x$, find the area of the region which contains the points which are closer to its centre than the sides.
Any ideas how to start.
I am having trouble figuring out the region itself
All ideas are appreciated
thanks
1
vote
1answer
48 views
Find $\int_0^{2}\int_0^{2}\left(x^2-2xy \right)\sqrt{1+4x^2+4y^2} \hspace{2mm} dydx$
I am having a tough time figuring this one
All help is appreciated
0
votes
3answers
55 views
Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $
I am having a tough time figuring this one out.
Any help will be appreciated. do we have to approximate, or can we actually find it
1
vote
1answer
39 views
Volumes of Revolution Washer Method
I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$.
...
1
vote
2answers
68 views
if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $?
I would be interest to show :
if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $ ?
my second question that's make me a problem is that :
what is :$ f^{-1}(\pi) $ ?
I would be ...
