Questions on the evaluation of definite and indefinite integrals
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7 views
Polynomials, integrals convergence
Let $P_n(x)= \frac{x^n(bx -a)^n}{n!}, \ \ \ a,b,n \in \mathbb{N}$.
Prove that $\int_0 ^{\pi}P_n(x) \sin xdx \rightarrow 0 \ \ \ \ $ and $ \ \ \ \ \int_0 ^r P_n(x)e^xdx \rightarrow 0$ $ \ \ \ \ \ \ (n ...
1
vote
1answer
37 views
Flow of $rot \overrightarrow{F}$
We've got vector field: $\overrightarrow{F} = \begin{bmatrix} yz\\x^3z\\e^z\end{bmatrix}$. I want to compute flow of $rot\overrightarrow{F} $($=curl \overrightarrow{F}$) through the area of the side ...
2
votes
2answers
48 views
real analysis and integral
Let $f$ be a continuous function on $[a, b]$ satisfying
$$\int_a^b f(x)g^\prime(x)\,\mathrm{d}x = 0$$
whenever $g$ is a continuously differentiable function on $[a, b]$ satisfying $g(a) =
g(b) = 0$.
...
2
votes
2answers
22 views
Problem involving the computation of the following integral
I was solving the past exam papers and stuck on the following problem:
Compute the integral $\displaystyle \oint_{C_1(0)} {e^{1/z}\over z} dz$,where $C_1(0)$ is the circle of radius $1$ around ...
0
votes
2answers
33 views
Checking $\displaystyle \int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$ for convergence
Given $\displaystyle\int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$, prove that it converges.
So of course, I said:
We have to calculate $\displaystyle \lim_{b \to \infty} ...
1
vote
1answer
22 views
$0$-th moment of product of gaussian and sinc function
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
6
votes
2answers
68 views
Computation of $\int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta$
Show that
$$\begin{align*} \forall x \in [-1,1]: \int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta &= c_n \tag{1} \\ \int_0^{\pi} \frac{\sin^{n+2} ...
1
vote
0answers
31 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
1
vote
2answers
45 views
How to show that these integrals converge?
What test do I use to show that the following integral converges?
If you could provide me with the process that leads to the answer that would really help.
$\displaystyle ...
0
votes
1answer
21 views
can a line integral of domain $C$ be negative when $C$ is the boundary of the region in the upper half plane? ($y>0$)
ok so here is the question:
evaluate $\oint \left(3y^2+e^{\cos x}\,dx\right) + \left(\sin y+5x^2\,dy\right)$, where $C$ is the boundary of the region in the upper half-plane ($y\ge 0$)between the ...
2
votes
1answer
26 views
Evaluate the following contour integral
I was solving old exam papers and I am stuck on the following question:
Evaluate the contour integral $\displaystyle \oint_{C} \frac{dz}{(\bar z-1)^2}$ where $C$ is the semi-circle $|z-1|=1, \Im ...
1
vote
1answer
40 views
Integration $\int \left(x-\frac{1}{2x} \right)^2\,dx $
$$\int\!\left(x-\frac{1}{2x} \right)^2\,dx $$
From U-substitution, I got $u=x-\frac{1}{2x},\quad \dfrac{du}{dx} =1+ \frac{1}{2x^2}$ , and $dx= 1+2x^2 du$
and in the end I come up with the answer to ...
3
votes
3answers
44 views
Integrate $\int {{{\left( {\cot x - \tan x} \right)}^2}dx} $
$\eqalign{
& \int {{{\left( {\cot x - \tan x} \right)}^2}dx} \cr
& = {\int {\left( {{{\cos x} \over {\sin x}} - {{\sin x} \over {\cos x}}} \right)} ^2}dx \cr
& = {\int {\left( ...
1
vote
1answer
50 views
Is the following differentiating under the integral sign correct?
Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial ...
5
votes
2answers
41 views
Evaluating the integral: $\int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx$
I am interested in evaluating the following integral:
$$
\int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx
$$
Using Matlab, Numerically it seems that the integral is convergent,
but I'm not sure ...
4
votes
1answer
34 views
$k$-th moment of product of gaussian and sinc
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
1
vote
0answers
25 views
Upper and lower integration inequality
I would like to learn how to prove that the following inequality holds.
Let $F$ be a bounded function on an interval $[a,b]$, so that there exists $B\geq 0$ such that $|f(x)| \leq B$ for every $x\in ...
0
votes
0answers
25 views
Stuck on a problem in multivariable calculus regarding flux help please!
I need help on this problem please! I tried doing it but I have been stuck on it for a while... some tutors couldnt even help me with this one.
Let $S$ be the portion of the sphere of radius a given ...
1
vote
3answers
42 views
Finding an area of the portion of a plane?
I need help with a problem I got in class today any help would be appreciated!
Find the area of the portion of the portion of the plane $6x+4y+3z=12$ that passes through the first octant where $x, ...
1
vote
1answer
26 views
We are to evaluate the problem at the given limit using pi and redicals in our answer as needed.
The Problem:
$$
\int\!\sin^5(4x)\,dx
$$
The formula that I used from the integration tables is:
$$
\int\!\sin^n(u)\,du
$$
My final answer is
$$
...
3
votes
3answers
31 views
Exponential integral question
How would I solve the following problem?
$$f(x)=\int\!\frac{4}{\sqrt{e^x}}\,dx$$
Using $u$ substitution I have set $u=e^x$ andd $du=e^x dx$
so would I have $$4\int\!\frac{du}{\sqrt{u}}$$
What would ...
2
votes
1answer
80 views
How to integrate $\cos\left(\sqrt{x^2 + y^2}\right)$
Could you help me solve this?
$$\iint_{M}\!\cos\left(\sqrt{x^2+y^2}\right)\,dxdy;$$
$M: \frac{\pi^2}{4}\leq x^2+y^2\leq 4\pi^2$
I know that the region would look like this and I need to solve it as ...
5
votes
3answers
80 views
Integral of $\cot^2 x$?
How do you find $\int \cot^2 x \, dx$? Please keep this at a calc AB level. Thanks!
5
votes
4answers
63 views
Integrate ${\sec 4x}$
How do I go about doing this? I try doing it by parts, but it seems to work out wrong:
$\eqalign{
& \int {\sec 4xdx} \cr
& u = \sec 4x \cr
& {{du} \over {dx}} = 4\sec 4x\tan 4x ...
1
vote
1answer
26 views
How to determine a function of 2 variables from its derivative?
Please even the slightest advice would help!
If I have a function $V$ made of 2 variables $x_1$ and $x_2$,
and its derivative $$\frac{dV}{dt} = \frac{dV}{dx_1}\frac{dx_1}{dt} + ...
3
votes
3answers
54 views
Proof for an integral identity
Is it true that $\int_0^A dx \int_0^B dy f(x) f(y) = 2 \int_0^A dx \int_0^x dy f(x) f(y)$ ? If so, can this be proved?
3
votes
4answers
47 views
What is the definite integral of…
$$\int^L_{-L} x \sin(\frac{\pi nx}{L})$$
I've seen something like this in Fourier theory, but I'm still not sure how to approach this integral. Wolfram Alpha gives me the answer, but no method. ...
7
votes
1answer
82 views
How to prove that $\int_0^{\infty} \log^2(x) e^{-kx}dx = \dfrac{\pi^2}{6k} + \dfrac{(\gamma+ \ln(k))^2}{k}$?
I was answering this question: $\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$ and in my answer, I encountered the integral
$$\int_0^{\infty} \log^2(x) e^{-kx}dx$$
which according to ...
6
votes
2answers
67 views
$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$
Is there any closed-form representation for the following integral?
$$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$
where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
0
votes
2answers
43 views
Describing Domain of Integration (Triple Integral)
I'm really struggling to go about starting the following problem:
This question concerns the integral,
$\int_{0}^{2}\int_0^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ ...
3
votes
3answers
67 views
Evaluate $\int x \cos x^2 dx$
Hay I have hit this in my book
Evaluate $\int x \cos x^2 dx$.
I got $x^2 \sin(x^2) / 2 $
But I used a online calculator to check it and it is giving me $\sin(x^2)/2 $
Where dose my X go?
4
votes
4answers
98 views
Integrate by parts: $\int \ln (2x + 1) \, dx$
$$\eqalign{
& \int \ln (2x + 1) \, dx \cr
& u = \ln (2x + 1) \cr
& v = x \cr
& {du \over dx} = {2 \over 2x + 1} \cr
& {dv \over dx} = 1 \cr
& \int \ln (2x ...
0
votes
1answer
34 views
Integral involving gaussian and triangle function
I would like to calculate the following integral:
$$\int_{-\infty}^{+\infty} \quad tri(x)\exp\left(-\frac{(x-x_0)^2}{a}\right)dx$$
Thanks!
1
vote
2answers
49 views
Convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$
Is there a closed form formula for the convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$, where $t>0$, i.e. the integral
$$\int_{-\infty}^\infty ...
0
votes
2answers
46 views
An integral problem?
How do you integrate $e^{e^x}$? I was able to get it down to du/(ln u) but I wasn't able to go further. Thanks!
2
votes
1answer
60 views
Evalute this integral using Green's Thereom
Let C be the boundary of the half-annulus
$$1\leq(x^2+y^2)\leq4$$ where $$x\le0$$
in the xy plane, traversed in the positive direction.
Evaluate : $ \displaystyle \int_{c}(7\cosh^3(7x)-2y^3) ...
3
votes
2answers
37 views
Integral of $\int^1_0\frac{dx}{\sqrt{x+3}-1}$
I want to solve this integral and need some directions.
$$\int^1_0\frac{dx}{\sqrt{x+3}-1}$$
I decided to call $x+3 = t^2 \rightarrow 2tdt = dx$ then : $$\int^1_0 \frac {2tdt}{t^2-1}$$ Now what should ...
2
votes
1answer
39 views
Integral of product of Bessel functions of the first kind
I would like to solve the integral:
$$\int_0^{+\infty}\quad rJ_n(ar)J_n(br)\quad dr$$
Is there any closed form for it?
Thanks!
2
votes
3answers
104 views
How to solve this integral for a hyperbolic bowl?
$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
2
votes
3answers
63 views
The indefinite integral $\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dx$
I`m trying to solve this integral and I did the following steps to solve it but don't know how to continue.
$$\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dx$$
$$\begin{align}\int \frac{\mathrm ...
2
votes
2answers
31 views
Integral of $\int \frac{\sin(x)dx}{3-\cos(x)}$
I am trying to solve this integral and I need your suggestions.
I don't know if its OK to set $3-\cos(x)$ as $t$ $\rightarrow dt = \sin(x)dx$ or just take $\cos(x)$ and set it as $t$
$$\int ...
1
vote
2answers
44 views
Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $
I'm working on the Fourier transform, but I don't know how to evaluate the integral:
$$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
0
votes
2answers
48 views
What's a better way to integrate this?
$$ \int \frac{1}{x^2 + z^2} dx $$
I tried substitution and also by parts. By parts is getting messy and I am not sure if I am getting the right answer. I am trying to figure out an easier way or the ...
-1
votes
0answers
15 views
Flux integrals, parameterization
let S be the cylinder x^2 + z^2 = 9 where -2 /ge y /le 2
parameterization: thi(u,v)= <3cosv, u, 3sinv> where -2 /ge y /le 2 and 0 /ge v /le 2pi
(thi is the symbol of I with the circle in the ...
0
votes
0answers
41 views
Evaluating a line integral directly
$F(x,y) = \dfrac{1}{x^2+ y^2}\langle -y,x\rangle$, and let $R$ be a circle of radius $a$, centered at the origin.
a) Why can't Green's theorem be used to evaluate $\int_R F \cdot ds$?
b) ...
0
votes
0answers
36 views
separating a variable from integral
In the following integral, I would like to separate $\alpha$ from rest of the equation. Can we solve the following equation for $\alpha$?
$$\large{\int_{0}^{a} \int_{0}^{2\pi} ...
-1
votes
1answer
31 views
Piecewise defined integration [closed]
Let
$$f_n(x) = \begin{cases}
0 & x < -\tfrac{1}{n} \\
\tfrac{n}{2} & -\tfrac{1}{n} \leq x \leq \tfrac{1}{n} \\
0 & x>\tfrac{1}{n} \\
\end{cases},$$
$n=1,2,3,\ldots$.
Let $F(x) = ...
17
votes
2answers
126 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
2
votes
4answers
69 views
$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$
I'm having trouble understanding how to apply the $\frac{d}{dx}$when taking the anti-derivative.
$$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$$
In class it was mentioned we'll end up taking ...
0
votes
1answer
38 views
$\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$
$$\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$$
Please help me in this integral, I've tried some substitutions, but nothing work.
Thanks in advance!
