Tagged Questions
0
votes
1answer
27 views
The sum of the integration of g and $g^{-1}$
Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto
$[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function.
Use geometric insight to visualize the ...
0
votes
3answers
57 views
How to solve these?
Inverse Trigonometric Functions
They are incomplete and I don't know how to complete them.
Who can help me?
1st
$$
\int\frac 1{ x \sqrt{x^{6} - 4}}dx
$$
I tried with:
$$u = x^3 $$
$$du= 3x^2dx$$
...
4
votes
1answer
62 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
0
votes
1answer
30 views
Prove $\int_2^\infty{\frac{\ln(t)}{t^{3/2}}},\mathrm{d}t$ converges
Show, using a comparison test, that $\displaystyle \int_2^\infty{\frac{\log{t}}{t^{\frac32}}}\mathrm{d}t$ converges.
All the answers I've tried shows it diverges, taking $\log{t} \le t^{1/2}$ and ...
2
votes
1answer
50 views
Simplification of an expression
How do I simplify the following expression?
$$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$
where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , $e ...
1
vote
3answers
89 views
$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?
Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities:
$$\|f*g\|_q\leq ...
2
votes
1answer
31 views
How to prove Chebyshev–Gauss quadrature integrate polynomial of degree less than $2n-1$ exactly
What I want to ask is mentioned in the title.
For example: how can we show that ...
1
vote
0answers
20 views
Gaussian quadrature with arbitrary weight function
In class, our professor told us how to evaluate the integral $\int_a^bw(x)f(x) dx$ by finding the Gaussian nodes $x_i$ and weight $w_i$ with weight function $w(x)=1$ (also known as Legendre ...
1
vote
2answers
40 views
Primitive function of $x^3 \sin x^2$
I'm trying to find the primitive function of $x^3 \sin x^2$, and I've come to a variable exchange ($t = x^2$) which led me to $\frac{1}{2} \int t \sin t dt$.
According to my text book, the primitive ...
3
votes
3answers
101 views
How do you integrate the following trigonometric function involving sin and cos?
How do you integrate the following functions:
$$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta
$$
...
2
votes
1answer
29 views
Evaluating a Gaussian Integral
How to prove that
$$\int_{\mathbb{R}^N}e^{-\langle Ax,x\rangle}\operatorname{dm}(x)=\left(\frac{\pi^N}{\det A}\right)^{\frac{1}{2}}$$
Where $A:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is a symmetric ...
4
votes
3answers
96 views
How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then
$$
\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0
$$
My attempt: I ...
1
vote
1answer
40 views
Integration Real Analysis
Let $E=\{1/n:n\in\mathbb{N}\}$ and consider the function on $[0,1]$ defined by $$f(x)=\begin{cases}\,1, &x\in E\\\,0,&\text{otherwise}\end{cases}.$$
Prove that $f$ is integrable on $[0,1]$ ...
2
votes
1answer
73 views
Can somebody provide an explanation to the formula of a one elementary integral?
Here is the formula:
$$
\int{\frac{dx}{x}} = \ln{|x|} + C
$$
In my textbook it is given without proof, so I have a little confusion here. From the definition of integral this equality must be true:
...
1
vote
1answer
50 views
Laplace equation and integral
$$ \int_0^{2\pi} \frac{1+3 \sin{\phi}}{a^2-2ar \cos(\theta - \phi) + r^2 } d\phi$$
Help me plz ... I have tried to solve this. but I still don't know.
1
vote
3answers
81 views
Integrating a school homework question.
Show that $$\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \frac{a\sqrt{3}+b-\pi}{6},$$ where $a$ and $b$ are constants to be found.
Answer is: $$\frac{24\sqrt3-48-\pi}{6}$$
Thank you in advance!
13
votes
3answers
218 views
Proving a trig infinite sum using integration
How can I prove the following using integration and elementary functions?
Prove that:
$$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$
$0 < \theta < 2\pi$
-1
votes
0answers
34 views
Please help me to solve this Henstock Integration [closed]
using cauchy criterion, show that the following functions are Henstock Integrable or not
a.$$f:[0,1]\to R \text{ where }f(x)=
\begin{cases}
1,\quad &x \text{ is rational}\,\\
0,&x \text{ is ...
-2
votes
2answers
43 views
prove $ e^{bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds $ [duplicate]
please How can I prove that
$$
e^{-bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds
$$
f non-negative measurable function
I would appreciate it enormously if ...
0
votes
1answer
46 views
prove that integral
prove that
$$-\int_0^t \ sgn(f({s})) \ d{s}=\int_0^t \ sgn(-f({s}))\ d{s}+2\int_0^t 1_{f({s}) =0}d{s}$$
with
$$ sgn(x) := \begin{cases}
-1 & \text{if } x =< 0, \\
1 & \text{if } x > ...
0
votes
2answers
60 views
prove Complicated Integral.
please How can I prove that
$$
e^{bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds
$$
f non-negative measurable function
best, Educ
4
votes
4answers
71 views
Evaluating Complex Integral.
I am trying to evaluate the following integrals:
$$\int\limits_{-\infty}^\infty \frac{x^2}{1+x^2+x^4}dx $$
$$\int\limits_{0}^\pi \frac{d\theta}{a\cos\theta+ b} \text{ where }0<a<b$$
My very ...
2
votes
2answers
37 views
Is there a need for another integration technique?
I'm being asked to calculate
$$I\triangleq\int_0^1\int_{e^{\large x}}^e{xe^y\over(\ln y)^2}\,dy\,dx\quad.$$
I got stuck on the indefinite inner one,
$$J\triangleq\int{e^ydy\over(\ln y)^2}\quad.$$
At ...
0
votes
1answer
27 views
Finding volume under surface and above a region
I'm asked to find $\underset{U}{\int}(x+y)^2\, dA$ where U is a region bounded by the lines
x = -1, x = 1, y = -1
... and by the curves
x=$y^2$ , y=1+$x^2$
Plot: http://d.pr/WYSg
I started out by ...
1
vote
3answers
49 views
Integration by parts, Reduction
I was able to complete part (a) easily by using integration by parts. I ended up getting:
$$I(n) = -\frac{1}{n} \cos x\cdot \sin^{n-1}x + \frac{n-1}{n}· I(n-2)$$
For question (b), When I ...
6
votes
2answers
62 views
Simple u-subsitution - Paradoxical Result
If I were to try and take $$\int{\mathrm{sin}(t)\mathrm{cos}(t)dt} $$ I would either take $u=\mathrm{sin}(t) $, yeilding a result of $\frac{1}{2} \mathrm{sin}^2(t) + C$, or I would take ...
0
votes
3answers
26 views
Integral question - $\int\frac{dx}{x\sqrt{\ln(x)}}$
This is the integral : $$\int\frac{dx}{x\sqrt{\ln(x)}}$$
This is what I did so far:
$$U=\frac1{\sqrt{\ln(x)}} \implies U'=-\frac{1}{2}{\frac{\ln(x)}{x}}$$
$$V'=\frac{1}{x} \implies V=\ln(x)$$
...
0
votes
1answer
22 views
Bounds of double integral given a circle and a line
Calculate the double integral of the area between the function $$x^2+y^2=25$$ and the line $$y=-x+5$$ in the first quadrant.
Now, I am unsure how to choose the bounds for y, I understand that the ...
1
vote
3answers
72 views
surface area of a sphere above a cylinder
I need to find the surface area of the sphere $x^2+y^2+z^2=4$ above the cone $z = \sqrt{x^2+y^2}$, but I'm not sure how. I know that the surface area of a surface can be calculated with the equation ...
1
vote
2answers
74 views
Integral Question $\int\frac{\sin^4(x)}{\cos^2(x)}\,dx$
What you are suggesting to do?
Convert $\sin^4(x)\Rightarrow (1-\cos^2(x))^2\,dx?$
$$ ∫\frac{\sin^4(x)}{\cos^2(x)}\,dx$$
Thanks!
1
vote
2answers
28 views
Is it true that $\int_1^ba^{\log_b x}dx> \log_eb$
Is it true that
$\int_1^ba^{\log_b x}dx> \log_eb$
$\forall a,b>0\ and\ b\not = 1$
1
vote
0answers
68 views
Please help on integration problem
I am having trouble integrating, especially since the question I am working on was not taught during the course.
The problem in general terms, integrate
$$\int_0^1{\frac{x^4}{\sqrt{1+x^3}}}dx$$
...
1
vote
2answers
84 views
Cylindrical coordinates where $z$ axis is not axis of symmetry.
I'm a little bit uncertain of how to set up the limits of integration when the axis of symmetry of the region is not centered at $z$ (this is for cylindrical coordinates). The region is bounded by ...
1
vote
0answers
41 views
Finding the average value of a function over region in $\mathbb{R}^3$.
I want to know if I set this multiple integral up correctly (I always mess them up!). I want to find the average value of $z$ over the region (call it $M$) bounded by $x^2+y^2+z^2=16$ and ...
1
vote
4answers
90 views
Integrating $\int{3\sin^2x\cos x \;dx}$
I am struggling with the following integral:
$$\int{3\sin^2x\cos x \;dx}$$
I have so far tried to solve this using every tool at my disposal, I have set $t = \sin x$ but I get an even harder ...
0
votes
2answers
68 views
What is wrong with this approach?
I am given a problem to find the surface area of the cylinder $z^2+y^2=9$ above the rectangle defined by the points $(0,0),(4,0),(0,2),(4,2)$.
Instead of trying to integrate on with any respect to ...
0
votes
0answers
11 views
Integral over bounded sets
Let $S_1$ and $S_2$ be bounded sets in $\Bbb R^n$; let $f:S \to \Bbb R$ be a bounded
function. Show that if f is integrable over $S_1$ and $S_2$, then f is integrable
over $S_1-S_2$, and
...
3
votes
1answer
64 views
How to find the integration of $\int \limits _{-\infty}^x e ^ \frac{-t^2}{2}{d}t$?
What is the value of the $\displaystyle \int \limits_{-\infty}^xe ^ {\large{-t^2/2}}dt$ ?
thank you for your time.
2
votes
3answers
118 views
Calculating the Fourier series of $x^{3}$
I was given as homework to calculate the Fourier series of $x^{3}$.
I know, in general, how to obtain the coefficients of the series using
integration with $$\sin(nx),\cos(nx)$$ multiplied by the ...
7
votes
2answers
162 views
Let $f:[a,b]\to\mathbb R$ be Riemann integrable and $f>0$. Prove that $\int_a^bf>0$. (No Measure theory) [closed]
Is the Riemann integral of a strictly positive function positive?
This is not a duplicate. I'm specifically interested in a proof not involving Measure Theory. The thread above uses the fact that $f$ ...
1
vote
4answers
43 views
Double Integral Gone Wrong
So, I have the (seemingly) algebraically innocuous double integral of $$ \iint \limits_R{1- {x^2 \over 4} -{y^2 \over 9}\space \mathrm {d}A} ; \space\mathrm {where}\space R =[-1,1]\times[-2,2] $$
I ...
9
votes
2answers
213 views
Let $f:[a,b]\to\mathbb R$ be Riemann integrable and $f>0$. Prove that $\int_a^bf>0$. (Without Measure theory)
The suggestion above is not relevent to my question.
I've been struggling with this for a while, and I have a couple of leads that kind of got me nowhere:
At first I thought that if $f$ is ...
2
votes
1answer
54 views
Proving that a weak solution of a BVP satisfies the boundary condition
I am given the smooth function $u$ which satisfies $\int_U
(\nabla u \cdot \nabla v +uv)\,dx = \int_U
fv\,dx$ for all functions $v$ in the Sobolev space $H^1(U)$, where $f\in ...
1
vote
1answer
108 views
Area between $f(x)= \sqrt{16-x^2}$ and $g(x)= e^{x^2}-2$
This is a problem to find the area enclosed by two function, $f(x)= \sqrt{16-x^2}$ and $g(x)= e^{x^2}-2$
...
4
votes
0answers
51 views
Integration by parts of a normalized function - [copied from Physics.SE]
By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that
...
1
vote
1answer
45 views
Find the center mass of right circular conic shell of base radius $a$ and height $h$ and constant density
We were assigned the question for homework:
Find the center of mass of a right circular conic shell, radius $a$, height $h$ and constant density $\rho$.
This is a multi-variable calculus class and we ...
0
votes
1answer
21 views
Outer measure defined by a continuous and bijective function
This problem is from K.T. Smith's Primer of Modern Analysis:
Let $\psi: \mathbb{R}^d \to \mathbb{R}^d$ be continuous and one-to-one on an open set $\Omega \subset \mathbb{R}^d$ and define $$\nu(A) ...
3
votes
3answers
92 views
How do I solve this problem with U-substitution?
$$\int \left(4-x(16-x^2)^{1/2}\right)\,dx $$
I learned today I could use U-substitution to before integrating, which makes it easier to integrate.
So I can make $U=16-x^2,\quad \dfrac{du}{dx} = ...
1
vote
1answer
59 views
Lebesgue Integral on Lebesgue measurable set satisfies Caratheodory condition
Let $f$ be a non-negative, measurable, and integrable over every compact set in $\Omega$, where $\Omega$ is an open set $\subset \mathbb{R}^d$.
For every Lebesgue measurable set $E$ (abbreviated as ...
3
votes
1answer
71 views
Please prove that $\lim\limits_{n \to \infty} \int_0^3 \sqrt{\sin(n/x)+x+1}\,dx$ exists and evaluate it.
Prove that $\displaystyle{\lim_{n \rightarrow \infty}} \int_0^3 \sqrt{\sin(n/x)+x+1}\,dx$ exists and evaluate it.
