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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0
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0answers
20 views

Wich way to do this integral?

Let $\delta(\phi) = \phi(0)$ be the dirac delta. I would like to compute $\int_{\mathbb{R}} h(x) \delta(\lambda x) dx$ 1) Since $\delta$ is an unit mass on $0$ $$\int_{\mathbb{R}} h(x) ...
3
votes
0answers
25 views

Origin of the Integral (Theory Behind It - How it came about)?

How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does ...
3
votes
1answer
36 views

Asymptotic behaviour of $\int_0^1 g(x)\exp(-nx)dx$ as $n\rightarrow\infty$

Let $g:(0,1]\rightarrow\mathbb{R}_+$ be an invertible monotonically non-increasing function that integrates to 1 and has g(1)=0, $g(0)=\infty$; eg. $g(x)=x^{-1/2}-1$ or $g(x)=ln(1/x)$. I believe it is ...
0
votes
1answer
28 views

Volume calculating using double integral

Here is my task: Calculate the volume under the surface $z=x^{2}-y^{2}$ over the region $(x^{2}+y^{2})^{3}=a^{2}x^{2}y^{2}$. Before solving this task, let's say that $z=x^{2}+y^{2}$ instead ...
1
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0answers
18 views

Analytic or numeric integration of Singular integral based on Bessel K0 and K1

thank you for reading this ! I need the following integral to be integrated from -1 to 1. It appears to have a singularity in -2/3. In string, Generated: ...
1
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4answers
40 views

When can I use the natural log to help solve an integral?

Why is it okay to do this: $\int \frac{1}{x-2}dx = \ln(x-2)$ but not this: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
6
votes
4answers
218 views

Integral involving a trig. term

I came across the following integral. $$ \int\frac{dx}{1+\sin x} $$ I have no idea how to solve it! I went for the obvious substitution of $u=1+\sin x$, but then you get an annoying $\cos x$ kicking ...
0
votes
5answers
85 views

What happens to an integral if the 'dt' disapears? (Integral with no dt)

I have been struggling with convolution yet have come up with my own method that involves cancelling the dt from the integral. I want to ask - If there is a term that is - ''Integral'' with limits ...
4
votes
1answer
44 views

Complex integral with Fourier

So, I've been scratching my head over this for the whole day. I'm trying to solve the following integral $$\int^\infty_{-\infty} \frac{e^{i \alpha (X-\xi)}}{\sqrt{\alpha^2+ \beta^2 }} \, d\alpha$$ ...
5
votes
2answers
180 views

Evaluating the indefinite integral $\int\log\!\left(x+\sqrt{x^2-1}\right)\!dx$

I came across the following integral, and I don't know how to solve it. $$ \int\log\left(x+\sqrt{x^2-1}\right)dx $$ I tried the "obvious" substitution of $x=\tan\theta$, which gives you: $$ ...
1
vote
2answers
29 views

Finding the integral of rational function of sines

$$ \int \frac{\sin x}{1+\sin x} \, \mathrm{d}x$$ How do I integrate this? I tried multiplying and dividing by $ (1- \sin x) $.
1
vote
3answers
75 views

Solve integrals using residue theorem? [on hold]

$$\int_{0}^{\pi}\frac{d\theta }{2+\cos\theta}$$ $$\int_{0}^{\infty}\frac{x }{(1+x)^6} dx$$ My problem is that I don't know how to start solving these integrals, or how to convert them into usual ...
0
votes
2answers
62 views

Integral of $x/(2x-1)$

I'm not sure how to do this, I'm also new to math.stackexchange so please excuse any novice mistakes. So anyways, here is a question I have on a summer assignment for Calculus BC (this is review from ...
1
vote
1answer
17 views

Convolution with one of the variables is mixed and the other continuous

Suppose $X$ and $Y$ are independent random variables with CDF $F$ and $G$ and nonnegative support. If $X$ has a point mass $p$ at $0$ and otherwise some "density" $f$ (that is, ...
0
votes
1answer
13 views

Integral identity for variable in integration limit

The following is an interesting integral identity: $H(t)=\int_0^tf(x,t)dx$, for $f(x,t)$ a sufficiently smooth function. Then, $H'(t)=f(t,t)+\int_0^tf_t(x,t)dx$. Why can't we use standard ...
0
votes
2answers
54 views

Convergence of $\int_0^\infty x^\alpha \cos e^x \, dx$

I tried to solve whether this integral is convergent or not and whether that convergence is conditional or absolute for a given $\alpha$. $$\int _0^{\infty }\:\:x^{\alpha \:}\cos\left(e^x\right)\, ...
0
votes
2answers
62 views

Integral of cos(1/x) dx

Is the following integral expression correct (neglecting the constant of integration)? $$ \int\cos\left(\frac{1}{x}\right)dx = x^2\sin\left(2x\right) $$ When I take the derivative, it returns to the ...
2
votes
2answers
45 views

Why is the integral starts from $0$?

Consider $$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$ It's a power series with a radius, $R=1$. at $x=1$ it converges. Hence, by Abel's thorem: $$\lim_{x\to 1^-} f(x) = ...
15
votes
1answer
199 views

Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$

Are we aware of an elementary way of proving that? $$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$ Of course, with the help of Mathematica it can be ...
0
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1answer
36 views

When calculating joint probabilities using double integrals…

When calculating joint probabilities using double integrals, do we use $dx\ dy$ or $dy\ dx$ ? I thought it was the former, but then my book abruptly changes to using $dy\ dx$ without an explanation ...
0
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0answers
21 views

Integration of convolution

I'm trying to solve the following equation $$\int\limits_{-\infty}^t \,(f\ast g)(t')dt'.$$ $f$ could be a kind of $\delta$-function: $f(t) = \delta(t)$ but should not be limited to be one. $g$ is ...
0
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0answers
11 views

Stoke's Theorem Application on Cylinder

This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW. My question is, referring to the answer provided, what closed surfaces are used in the proof ...
1
vote
0answers
37 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
1
vote
0answers
36 views

How to evaluate the integral ${\displaystyle\int_0^{m_1} \int_0^{m_2} }\frac{dx' dy'}{[(x-x')^2+(y-y')^2+25]^{\frac{3}{2}}} $

How to evaluate the integral $$\int_0^{m_1} \int_0^{m_2} \frac{dx' dy'}{[(x-x')^2+(y-y')^2+25]^{\frac{3}{2}}} $$
0
votes
1answer
28 views

Interchange Order of Integrals

Can someone explain the last step in this process. Specifically, how do you get the new limits of integration? Expected Value Definition: $E[Y] = \int_0^\infty{P\{Y \ge y\} \, dy}$ Expand: $E[Y] = ...
2
votes
3answers
77 views

Integrate $\displaystyle \int \sin(\sqrt{at})dt$

Integrate $\displaystyle \int \sin(\sqrt{at})dt$ Here is what I tried. Let $u=\sqrt{at}$, then $\displaystyle\ du=\frac{a}{2\sqrt{at}}dt=\frac{a}{2u}dt\implies \frac{2udu}{a}=dt.$ So by subsitution, ...
0
votes
1answer
18 views

Derive $E(X^k)$ I need help with the substitution piece.

If $X\sim\mathrm{WEI}(\theta,\beta)$, derive $E(X^k)$ assuming $k > -\beta$. Note that $X\sim\mathrm{WEI}(\theta,\beta)=\dfrac{\beta}{\theta^\beta}x^{\beta -1}e^{-(x/\theta)^\beta}$ I know to ...
8
votes
1answer
135 views

How to find $I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$

How can I find this integral $$I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$$
0
votes
3answers
79 views

Evaluate the following integration below [on hold]

Evaluate the following integration $$\int_{0}^{\infty }\frac{x^2}{x^6 +1}dx$$ help guys please, I tried but I got nothing.
0
votes
1answer
106 views

Integrating $\int{\frac{x^2}{1+x^5}dx}$ [on hold]

I just encountered the following integral $$\int{\frac{x^2}{1+x^5}dx}$$ At first it appeared to be simple, but I don't know how to solve it. Please share any ideas.
2
votes
2answers
46 views

Definite Integration, keep getting wrong answer.

Correct to 4 significant figures $$\int_{1}^{2}{\csc^24tdt}$$ Done this multiple times now and can't seem to get the answer at the back of the book. Here's my attempt: ...
1
vote
0answers
30 views

Integral of $|\cos(ax))|\times e^{-x^2/b}$

I can compute the following integral very easily ($a$ and $b$ are real and positive): $$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = ...
-1
votes
3answers
21 views

Area bound by two curves. [on hold]

What is the area of the region bounded by the curves $y= 2x^2+7x$ and $y= 2/x$ between $x= 1$ and $x= 3$? Thank you!
6
votes
2answers
48 views

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

closed form for : $$\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{\mathrm{dx}}{(b^2+x^2)}$$ where $\beta$ is beta function I tried with the definition of beta and i got ...
0
votes
0answers
19 views

Showing certain sum as a Riemann-Stieltjes integral

Let $e(\beta) = e^{2 \pi i \beta}$. I am reading an article, where the author defines the following sum $$ S(N) = \sum_{0 \leq x \leq N, x \equiv g (mod \ q)} \Lambda(x) e(f(x) \alpha), $$ where $f$ ...
2
votes
3answers
96 views

Simplest way to integrate this expression : $\int_{-\infty}^{+\infty} e^{-x^2/2} dx$ [duplicate]

I'm toying around with statistics and calculus for a project of mine and I'm trying to find the simplest/fastest way to integrate this formula : $$\int_{-\infty}^{+\infty} e^{-x^2/2} dx$$ I do not ...
-1
votes
1answer
47 views

Proving that the integrals of two functions are the same if they are equal everywhere except a point [on hold]

Let $f(x)$ and $g(x)$ be integrable functions over $[a,b]$ and let $∂$ be a point on $[a,b]$. If $f(x) = g(x)$ for all $x≠∂$, then $$\int_a^b f(x)dx=\int_a^b g(x)dx$$
1
vote
3answers
52 views

How do i find this : $\int \frac{1}{(x+a) \sqrt{x+b}}\ dx$, where $a > b > 0$?

Is there someone show me how do I find : $$\int \frac{1}{(x+a) \sqrt{x+b}}\ dx$$, where $$a > b > 0$$ ? I tried to make it as sum of fraction to be easier but sorry i didn't up Thank you for ...
0
votes
0answers
16 views

Implicit numerical integration: error bound

Suppose I'm solving this equation numerically with a time step $h$: $$x''(t) = f(x)$$ Discretizing it and using implicit integration: $$x^{n+1} - 2x^n + x^{n-1} = h^2f( x^{n+1})$$ $x^{n-1}$ and ...
3
votes
7answers
128 views

Evaluating the indefinite integral $\int e^{-x^2}\,\mathrm{d}x$ [on hold]

In my book, it is said that $$\int e^{-x^2} \, \mathrm{d}x$$ cannot be solved by the method of inspection. It then turned to method of substitution as a new topic. I am not able to solve this ...
6
votes
3answers
76 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I supposed that : $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx .$$ For $a=2$ I have got : $2\pi$ I think the result will be : ...
0
votes
1answer
60 views

Calculus 2 - $\int(\sqrt{72+36x^2}dx$

I have done this problem several times and this is the only answer i ever come to. My schools webwork gives me incorrect for my answer (answer is not simplified but it should be accepted in this ...
-2
votes
1answer
51 views

Riemann integrability of the function that is equal to $0$ only at $1,1/2,1/3,…$ and $1$ otherwise [on hold]

How can I prove that the following function is Riemann Integrable on [0,1] using the criterion $U(P,f)-L(P,f)< \varepsilon$ \begin{equation} f(x)=\begin{cases} 0 & \text{if } x=1, 1/2, 1/3, ...
1
vote
1answer
27 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
1
vote
2answers
71 views

How do I evaluate this integral $I = \int_{0}^{2 \pi} \ln (\sin x +\sqrt{1+\sin^2 x}) dx$?

I used some variables change to evaluate this integral but i'm not succeed may I have some wrong step as trigono-transformation.Then Is there some one who can show me how do evaluate this : $$I = ...
0
votes
2answers
57 views

Find the area of the entire region that lies between $r=1+\sin\theta; r=1+\cos\theta$

I have to find the area of the region that lies between the curves $r=1+\sin\theta; r=1+\cos\theta$ . The answer the book gave was $\frac {3\pi}{2}-2\sqrt{2}$ . I tried generating the curve for ...
0
votes
1answer
19 views

Derivative of an integral over a varying domain

Consider the function $$H(\alpha) = \int_{\Omega(\alpha)} h(\alpha,x) dx,$$ where $\alpha\in\mathbb{R}$ and $\Omega(\alpha)\subset\mathbb{R}^2$ is a domain that varies continuously with $\alpha$. Is ...
1
vote
1answer
19 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
9
votes
3answers
97 views

What's happening at $a=-1$ in $\int x^a dx$? [duplicate]

If we take the right limit $$\lim_{a\to-1}\int x^a dx=\lim_{a\to-1}\frac{x^{a+1}}{a+1}=+\infty$$ but on the other hand $$\int\lim_{a\to-1} x^a dx=\ln x$$ I'm aware you can't just commute the ...
4
votes
1answer
68 views

Area under tangent to a curve.

The tangent to the graph of the function $y=f(x)$ at the point with abscissa $x=a$ forms with the line $x$-axis an angle $\frac{\pi}{6}$ and at the point with abscissa $x=b$ an angle of ...