Tagged Questions

Questions on the evaluation of definite and indefinite integrals

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0answers
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 ...
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 ...
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 $g(a) = g(b) = 0$. ...
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 ...
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} ...
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 ...
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} ...
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 ...
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 ...
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 ...
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 ...
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 ...
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( ...
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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  ...
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 ...
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 ...
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!
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 ...
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} + ...
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?
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. ...
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 ...
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, ...
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\ ...
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?
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 ...
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!
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 ...
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!
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) ...
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 ...
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!
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)
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 ...
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 ...
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: $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx}$$
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 ...
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 ...
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) ...
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} ...
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) = ...
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: $$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
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 ...
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!
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