Questions on the evaluation of definite and indefinite integrals

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