Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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10 views

Is there any way to solve integral of sqrt(8-x^2) without using sin or cos formulas?

I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace x with ...
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3answers
21 views

intergate $\int \frac{x}{(x^2-3x+17)^2}$

$$\int \frac{x}{(x^2-3x+17)^2}$$ $$\int \frac{x}{(x^2-3x+17)^2}=\int \frac{x}{(x-\frac{3}{2})^2+\frac{59}{4}}$$ $u=x-\frac{3}{2}$ $du=dx$ $$\int \frac{u+\frac{3}{2}}{(u)^2+\frac{59}{4}}$$ How ...
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1answer
67 views

Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
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3answers
30 views

Simple integration just learning this application

I want to integrate a function as $f(x)=\sin^{-1}x$. What should be the proper method of doing it?
2
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3answers
46 views

integrate $\int \frac{dx}{(9+x^2)^2}$

$$\int \frac{dx}{(9+x^2)^2}$$ $x=3\tan\theta$ $dx=\frac{3}{\cos^2\theta}d\theta$ $$\int\frac{\frac{3}{\cos^2\theta}}{(9[1+\tan^2\theta])^2} \,d\theta = ...
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0answers
34 views

Solution of this definite integral?

I want to find the expression for the following integral $$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$ I have tried deriving wrt $k$, transforming into an integral over the whole real line... with ...
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0answers
8 views

Cdf of truncated distribution

Let $X$ be a random variable with density $f_x$ and distribution function $F_x$. Define the interval $I = (a,b)$. Given that we know these and the inverse distribution function $F^{-1}_x$, how can we ...
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29 views

Integrating to find Action in Quantum Field Theory

I am struggling to show: $\int_{w=0}^{w'} \int_{r=2M}^{2(M-w)} \frac{-drdw}{1-\sqrt{\frac{2(M-w)}{r}-\frac{Q^2}{r^2}}}=2\pi[{2w'(M-\frac{w'}{2})-(M-w')\sqrt{(M-w')^2-Q^2)}+M\sqrt{M^2-Q^2}}]\\$ A ...
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2answers
26 views

trigo substitution and identites?

When I use trigo substitution to solve an integral I get an expression like that: $$\frac{1}{4}\tan\left(\arcsin\left(\frac{x-2}{2}\right)\right)+C$$ How can I simplify it?
3
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3answers
59 views

$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$

$$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$$ where V is the unit sphere. No information is given about n but I assume it is an integral. All I could think to do was to convert to spherical ...
3
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1answer
22 views

Intuition behind: Integral operator as generalization of matrix multiplication

So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications. In ...
2
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2answers
63 views

integrate $\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$

$$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$$ $$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx=\int \frac{3\left(\frac{16}{9}-x^2\right)^{\frac{3}{2}}}{x^6}dx$$ $x=\frac{4}{3}\sin\theta$ ...
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1answer
50 views

Challenge in trignometry and integration [on hold]

Can anyone prove how the two equations are equal? Thanks $$=\frac1\pi \int_0^{2\pi} f(x) \left\{\frac12+\sum_{n=1}^N \cos [n(t-x)] \right\} \, dx$$ $$=\frac1{2\pi} \int_0^{2\pi} f(x) ...
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1answer
38 views

Evaluate $\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x) \sin x} $

Evaluate $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx \:\:\: n \in \mathbb{N}$$ $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx + ...
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1answer
24 views

Approximating a Riemann integrable function using a continuous function

Let $f$ be Riemann integrable on $[a,b]$. Show that for every $ε > 0$, there is a continuous function $g$ on $[a,b]$ such that $$\int_a^b |f(x)−g(x)|\mathrm dx < ε. $$
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3answers
380 views

Integration with a constant “a”: $ \int_0^a \frac1{\sqrt {a^2-x^2}} dx $

Find the exact value of $$ \int_{0}^{a} \frac{1}{\sqrt {a^2-x^2}} dx $$ Where, $a$ is a positive constant Hi, guys can give me tips to solve this ? Should we use like u substitution?
2
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0answers
30 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
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0answers
14 views

Is there any trick for evaluate this integral?

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
1
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1answer
27 views

Verify that $Γ(x)$ = $(x − 1)Γ(x − 1)$ for all $x > 1$.

$Γ(x)$ = $\int_0^{∞} e^{-t}t^{x-1}dt$ Plugging $(x-1)$ into this equation, I get $Γ(x-1)$ = $\int_0^{∞} e^{-t}t^{x-2}dt$ Integrating by parts, I eventually end up with $-e^{-t}t^{x-1}]_0^∞$ + ...
3
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1answer
30 views

Solve the differential equation : $0.5 \frac{dy}{dx}=4.9-0.1y^2$

The question is to solve the differential equation : $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ What I have attempted: $$0.5 \frac{dy}{dx}=4.9-0.1y^2$$ $$ \frac{dy}{dx} = \frac{4.9-0.1y^2}{0.5} ...
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1answer
21 views

Find the volume of the solid generated by the region [on hold]

Find the volume of the solid that is generated when the region enclosed by $ y = \cosh 2x, y = \sinh 2x, x = 0, $ and $ x = 5 $ is revolved around the x-axis.
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0answers
37 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
2
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1answer
60 views

Find $\int\limits^{\infty}_{0}\int\limits^{\infty}_{0}{\frac{1}{(x+y)^{3/2}}\exp\left\{-\frac{a^2}{2(x+y)}\right\}}\,dy\,dx$.

In my posterior probability computation, I got the following integration and I could not figure it out. ...
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0answers
23 views

Piecewise function evaluation, using integration. [on hold]

So i have this question and im completely lost. can someone help me please! I tried the first question, but its not correct, my answer was $g(-3) = 0$ and so it $g(3) = 0$ but apparently, i did ...
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2answers
25 views

Is it true in general that $\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r$?

If I have that $X$ is a random variable, for $\epsilon > 0$, and $r \geq 1$, is it true that: $$\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r.$$? If so, is there a reason why? ...
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0answers
33 views

Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$

$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$ for $t\in\Bbb{N}$ How might one represent $a,b$ in terms of $t$? (Note that $a,b\in \Bbb{N}$) If possible, could one also ...
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2answers
35 views

Differential Equations $ v \frac{dv}{dx} = -g \frac{a^2}{x^2}$

Question: A particle is projected vertically upwards from the Earth's surface. Its distance $x$ from the centre of the Earth is connected with its upwards speed $v$ by the differential ...
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3answers
64 views

Without computing, is the integral of $\int_0^1 t(t-1)(t-2)\,dt$ positive or negative? [on hold]

I have to graph the function, but I don't think I'm doing it right. Here is a picture of it Sorry, this is my first time using this site and I don't know how to use MathJax yet.
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0answers
15 views

Need help with $\int\mathrm{exp}[-C(\frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1)^S ] \mathrm{dx}$ for a statistical mechanics problem

Can someone help solving this integral?: $$\int\limits_{-\frac{1}{2}}^{0} \mathrm{exp}\left[-C \left( \frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1 \right)^S \right] \mathrm{dx}$$ The ...
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3answers
32 views

Integrate the following equation. (exponential function)

Integrate $$\frac{e^x -2}{e^{x/2}}$$ This is my calculation: but it is wrong....
3
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0answers
20 views

Riesz-Type Representation Theorems for Convex Functionals

It is well known that any positive linear functional $L$ on the spase $C_c([a,b])$ of functions continuous on an interval $[a,b]$ with compact support can be written as \begin{align*} ...
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0answers
29 views

Problem with $\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$ (by residues) [duplicate]

I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$ Teachers teached me that I can solve the integral $$\int_{0}^{\infty} ...
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3answers
45 views

Area of a rectangle within a curve

The cargo space of a bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. It is shaped like a parabola with equation ${{1\over 4}x^2, - 6 \le ...
1
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1answer
28 views

Combining two results from partial integration

I have a set of two PDEs: $$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$ $$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$ These can be combined into a wave equation of ...
1
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1answer
19 views

Computing a line integral where the curve is in polar coordinates

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(y,x)$ and $C$ is the curve given by $r=1+\theta$ for $\theta \in [0,2\pi]$ My Attempt Am I correct in saying that $F$ is a conservative vector field ...
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1answer
40 views

solving $\int_0^{\pi/2} sin^2\theta \sqrt{1-k^2sin^2 \theta}d\theta $

I have the following integration to solve. $$f(k) = \int_0^{\pi/2} sin^2\theta \sqrt{1-k^2sin^2 \theta}d\theta $$ assuming $sin\theta = t$ which results $d\theta = \frac{dt}{\sqrt{1-t^2}}$ and when ...
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2answers
50 views

How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$

How to integrate $$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$ where $a>0$ The real problem is this integral $$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) ...
1
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2answers
74 views

Find $\lim_{n \rightarrow \infty}\frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$

Find: $$\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$$ The sequence $\frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ ...
3
votes
3answers
54 views

Find $\lim_{n \rightarrow \infty} \int_0^n (1+ \frac{x}{n})^{n+1} \exp(-2x) \, dx$

Find: $$\lim_{n \rightarrow \infty} \int_0^n \left(1+ \frac{x}{n}\right)^{n+1} \exp(-2x) \, dx$$ The sequence $\left(1+ \frac{x}{n}\right)^{n+1} \exp{(-2x)}$ converges pointwise to $\exp{(-x)}$. So ...
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1answer
17 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
1
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1answer
45 views

Weird indefinite integral homework questions

I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems ...
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2answers
33 views

Evaluate the inverse trigonometric integral

Evaluate the integral:$\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx$ On applying the property $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$ I dont seem to reach any where
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1answer
27 views

Evaluating the integral of a sine function

I am having some trouble with part (b) and part (c) of this: (b) I know that I have to differentiate it and I get $\cos (\frac{\pi}{x})$ and by using the definite integral I get $\cos (\pi n)-\cos ...
2
votes
1answer
54 views

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$? I calculated the integrating factor to be $e^x$. Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $ ...
0
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0answers
24 views

Definition of integrability for sequences

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
0
votes
1answer
26 views

Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
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0answers
33 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
1
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2answers
30 views

Find volume of these solids using integration

a) The $(x>0, y< -1)$ region of the curve $y= -\frac{1}{x}$ rotated about the $y$-axis. The instructions say that one should use the formula: $V = \int 2πxf(x) dx$ I used another method and ...
2
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0answers
87 views

How do I evaluate this special type of integral

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
0
votes
0answers
38 views

Finding the integral of a 1/variable*radical function

I'm trying to find the integral of $$\int\frac{1}{x* (\sqrt{4x^4 - 9})}$$ Attempt: I assumed that the integral would be some sort of inverse trigonometric function. Because of this, I did the ...