Questions on the evaluation of definite and indefinite integrals
1
vote
2answers
48 views
Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution
Consider this integral:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$$
How would you compute it?
I already solved this problem this way:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \left( ...
3
votes
2answers
84 views
$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$ , $\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$
Please help me integrate
$$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$$
and
$$\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$$
I've tried the standard $u = \tan \frac{x}{2}$ substitution but it looks ...
4
votes
3answers
90 views
Integrating left to right versus right to left.
OK, I understand that when integration is done left to right with respect to x increasing left to right (dx is positive), that the answer is positive, and vice versa when integrating right to left. ...
2
votes
1answer
33 views
Evaluating Complex Line Integrals
Calculate $\int_{\gamma}\frac{\Re(z)}{z-\frac{1}{2}}dz$ and $\int_{\gamma}\frac{\Im(z)}{z-\frac{1}{2}}dz$ when $\gamma$: $|z|=1$ is positively oriented.
This is what I have tried to do, starting ...
1
vote
0answers
46 views
$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}$
I'm stuck on my last exercise. Could you help?
$$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)} \ dx$$
3
votes
0answers
30 views
What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?
I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$.
...
1
vote
3answers
43 views
Integration of a rational function from +/- infinity
I am trying to calculate the integral
$$\int_{-\infty}^{\infty}{\frac{a+x}{b^2 + (a+x)^2}\frac{1}{1+c(a-x)^2}}dx$$
where $\{a, b, c\}\in \mathbb{R}$. I have looked in a table of integrals for ...
0
votes
1answer
45 views
$ \int_{0}^{\infty}{\dfrac{\cos(ax)}{(x^2 + 1)^2}dx} $
I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct.
$$ \int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx} $$
For this one, ...
2
votes
1answer
49 views
How to place a limit that it's inside the integral, outside.
I did this:
$$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow ...
2
votes
0answers
50 views
Double Integral Homework Problem
Here's the problem statement of the question which I am stuck on:
Let $R_{1}$ denote the rectangle $[0, 5] \times [-4, 4]$, $R_{2}$ the rectangle $[0, 5] \times [0, 4]$, and $R_{3}$ the rectangle ...
2
votes
2answers
31 views
Quadrature formula
How can we find a quadrature formula $\int_{-1}^1 f(x) dx=c \displaystyle \sum_{i=0}^{2}f(x_i)$ that is exact for all quadratic polynomials?
Thanks for help.
0
votes
2answers
77 views
Integral question help me?
I am in the middle of solving a diff equation and I have to solve $\int\dfrac{1}{\cos x}e^{\tan x}\,dx$? I am thinking about integration by parts but it is a very long way.is there another short way? ...
0
votes
1answer
22 views
Integral, set and parametric representation
I am to compute the following: $\displaystyle\iiint\limits_V 1\, dx\, dy\, dz$,
where $V= \{{(x,y,z) \in \mathbb R^3 : (x-z)^2 +4y^2 < (1-z)^2} \text{ and } 0<z<1\}.$
Does anyone have idea ...
2
votes
3answers
40 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
3
votes
4answers
76 views
Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?
How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
1
vote
4answers
59 views
Integral of $\int \frac{x^4+2x+4}{x^4-1}dx$ [duplicate]
I am trying to solve this integral and I need your suggestions.
$$\int \frac{x^4+2x+4}{x^4-1}dx$$
Thanks
1
vote
3answers
51 views
How to evaluate $\int_1^\infty \frac{1}{x}-\frac{1}{x+1}~dx$
It's a very simple question but it confuses me. How do I evaluate
$$
\int_1^\infty \frac{1}{x}-\frac{1}{x+1}~dx
$$
without splitting? And why can't I split it?
7
votes
1answer
139 views
Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$
Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that
$$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$
Prove that
$$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$
Where should ...
0
votes
1answer
54 views
Theorem or just a change of varibles?
I have a formula in my text:
$$\int \int_{S} F \cdot n dA= \int \int_{w} F(G(u,v)) \cdot (dG_{u}\times dG_{v}) du dv$$
I am really lazy and hate remembering formulas to me this looks like a ...
6
votes
3answers
97 views
Integral of $\int^1_0 \frac{dx}{1+e^{2x}}$
I am trying to solve this integral and I need your suggestions.
I think about taking $1+e^{2x}$ and setting it as $t$, but I don't know how to continue now.
$$\int^1_0 \frac{dx}{1+e^{2x}}$$
Thanks!
2
votes
1answer
82 views
Limit of a continued fraction
Given the continued fraction:
$$f(x,N)=\left[2,3,4,...N,x\right]$$
$$f(x,N)=\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{...+\cfrac{1}{x}}}}}$$
is it possible to find an expression for the integral:
...
1
vote
2answers
39 views
Integral of $\int(4-2x)^\frac{1}{3}dx$
I solved this integral then I did $\frac{d}{dx}$ of $F(x)$ and saw that its not the same, so I did wrong in my integration process.
$$\int(4-2x)^\frac{1}{3}dx$$
What I did is $$F(x) ...
0
votes
5answers
85 views
$\int^1_0 \frac{xdx}{x^2+2x+1}$
I need some suggestion how to solve this integral.
$$\int^1_0 \frac{xdx}{x^2+2x+1}$$
I think about to do the following step :
$$\frac{1}{2}\int^1_0\frac{2x+2-2dx}{x^2+2x+1}$$$$ t=x^2+2x+1 \rightarrow ...
5
votes
2answers
58 views
Integral of fractional expression $\int^3_0 \frac{dx}{1+\sqrt{x+1}}$
I want to solve this integral and think about call $\sqrt{x+1} = t \rightarrow t^2 = x+1$
$$\int^3_0 \frac{dx}{1+\sqrt{x+1}}$$
Now the integral is : $$\int^3_0 \frac{2tdt}{1+t}$$ now I need your ...
3
votes
1answer
50 views
Calculus II, Curve length question.
Find the length of the curve
$x= \int_0^y\sqrt{\sec ^4(3 t)-1}dt, \quad 0\le y\le 9$
A bit stumped, without the 'y' in the upper limit it'd make a lot more sense to me.
Advice or solutions with ...
15
votes
1answer
153 views
Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick
I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several ...
17
votes
2answers
177 views
+100
An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$
I need to calculate the following integral:
$$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$
where
$$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$
...
1
vote
1answer
53 views
Improper integral sin(x)/x converges absolutely, conditionaly or diverges?
$$\int_1^{\infty}\frac{\sin x}{x}dx$$
$$u=\frac{1}{x}$$
$$du=-\frac{1}{x^2}dx$$
$$dv=\sin xdx$$
$$v=-\cos x$$
$$\int_1^{\infty}\frac{\sin x}{x}dx=\frac{1}{x}(-\cos x)-\int_1^{\infty}\frac{\cos ...
3
votes
2answers
64 views
Integration involving roots
$$\int\frac{dx}{(1+x^\frac{1}{4})x^\frac{1}{2}}$$
This is my work:
$$u^4=x$$
$$4u^3=dx$$
$$\int\frac{4u^3du}{(1+u)u^2}=\int\frac{4u^3du}{(1+u)u^2}=-4(1+x^\frac{1}{4})^{-1}+2(1+x^\frac{1}{4})^{-2}+C$$
...
4
votes
1answer
53 views
The geometric interpretation [duplicate]
In the course of mathematical analysis, there was one problem that i excited to know more about it:
What is the geometric interpretation of
$$ \int_a^b f(x)\,d(\alpha(x)) $$
and $\alpha(x)$ is ...
14
votes
1answer
139 views
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
Please help me to find a closed form for the following integral:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
2
votes
0answers
63 views
A photon in expanding Universe (a snail on a tree)
I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
0
votes
1answer
22 views
How to solve expectation / first moment of Gaussian Integral?
How can I solve the following integral?
$E[I] = \int_{I=0}^\infty I \frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{(\mu-f-I)^2}{2\sigma^2}) dI$
The result is supposed to be:
$\sigma[\frac{\mu - ...
1
vote
2answers
35 views
Proving that $\frac{\sigma_{n-1}}{\omega_n} = n$ in $\mathbb{R}^n$
If $\sigma_{n-1}$ was the surface area of the unit sphere in $\mathbb{R}^n$ and $w_{n}$ was the area of the unit ball in $\mathbb{R}^n$, my lecture notes prove that $$\frac{\sigma_{n-1}}{\omega_n} = ...
3
votes
2answers
72 views
Integral of $ \int_{-1}^{1} \frac{x^4}{x^2+1}\,dx $
Any suggestions how to solve it? by parts?
$$ \int_{-1}^{1} \frac{x^4}{x^2+1}dx$$
Thanks!
2
votes
2answers
41 views
What's the meaning of oscI(fn-f)?
I haven't seen the form of osc$_I(f_n-f)$.I expect your explanation.
3
votes
2answers
51 views
Using area=$\int_{y = a}^{y = b} x \, dy $ find the following shaded area:
So I did:
$$\eqalign{
& y = {1 \over x^2} \cr
& {x^2} = {1 \over y} \cr
& x = y^{ - {1 \over 2}} \cr
& \int_a^b x \, dy = \left[ 2{y^{1 \over 2}} \right]_{1 \over ...
1
vote
1answer
37 views
Solving an integral equation: $y(x) = 16+\int_0^x 2t \sqrt{y(t)} dt$
How do I solve the following integral equation?
$$y(x) = 16+\int_0^x 2t \sqrt{y(t)} dt$$
7
votes
3answers
76 views
Arc length of logarithm function
I need to find the length of $y = \ln(x)$ (natural logarithm) from $x=\sqrt3$ to $x=\sqrt8$.
So, if I am not mistake, the length should be $$\int^\sqrt8_\sqrt3\sqrt{1+\frac{1}{x^2}}dx$$
I am having ...
2
votes
2answers
33 views
I need help to evaluate this integral with a constant $c$
Find the value of the constant $c$ for which the integral converges, and evaluate the integral:
$$\int_0^\infty \left(\frac{9x}{x^2+1}-\frac{9c}{2x+1}\right)dx$$
5
votes
1answer
89 views
A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$
Is there a special trick to calculate this integral?
$$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$
for $\lambda>0$.
3
votes
2answers
94 views
Integration of radial functions?
Let $f(|x|)$ be a integrable radial function in $\mathbb{R}^n$ ($|\cdot|$ denotes the euclidean norm as in convention). The following identity is used to simplify computations
...
3
votes
1answer
31 views
convergence of an integral ( with an inner integral)
I need to figure out for which values of $p \in R $ does the following integral converge?
$$\int_0^{1} \frac{x^p}{\int_0^{x}\ln(1 + \sin(t) + t)dt} {dx} $$
Please note that I don't have to ...
0
votes
0answers
72 views
Simplifying $\frac{1}{n}\sum_{k=1}^n f(\frac{1}{k})$
Suppose that $$\displaystyle \forall x\in \mathbb{R}_+^* \quad f(x)=\frac{x^2-1}{4}-\frac{\ln(x)}{2}.$$
How can I simplify this: $$I(n)=\frac{1}{n}\sum_{k=1}^n f\left(\frac{1}{k}\right)$$
and prove ...
0
votes
2answers
26 views
Finding the area between two curves and the x-axis
The two curves $y = 3\sqrt{4-x}$ and $y = 3\sqrt{x-2}$ are given. The task is to find the area enclosed by the curves and the x-axis.
As far as I see, the root of the first function is $x = 4$ and ...
1
vote
2answers
53 views
Explanation of why $\int \sin^2 x\cos^2 x\; dx = 1/3 \sin^3 x - 2/5 \sin^5 x + 1/7 \sin^7 x +c$
verify the solution
$$\int \sin^2 x\cos^5 x\; dx = 1/3 \sin^3 x - 2/5 \sin^5 x + 1/7 \sin^7 x +c$$
I have hit this in my book and can't work it out. Does anyone have any ideas or a walk-through ...
0
votes
1answer
23 views
Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$
I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$
...
14
votes
1answer
166 views
$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$
I need help with calculating this integral:
$$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$
where ...
7
votes
2answers
111 views
Proving that $\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}$ by induction
I need to prove
$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$
I've seen other demonstrations of this, but they use some identities that I don't understand. ...
0
votes
1answer
53 views
Does anyone recognize the function: ${ F }_{ \delta }(x)=\frac { 1 }{ 2\delta } \int _{ -\delta }^{ \delta }{ f(x+t)dt } $
I was given as homework a question related with this "smoothing" function:
${ F }_{ \delta }(x)=\frac { 1 }{ 2\delta } \int _{ -\delta }^{ \delta }{ f(x+t)dt } $
Where $f(x)$ is a continuous ...
