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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1answer
139 views

Help evaluating $\int_{-1/2}^{1/2}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$

$$\int_{-1/2}^{1/2}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)df$$ I've tried simplifying the integrand, but I can't get to a point where I can evaluate the integral. I know ...
-1
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1answer
72 views

Calculate $\int^{n+1}_1 \frac{(\{x\})^{[x]}}{[x]}dx$ where $n\in\mathbb{N}$, $[x]$ is the integer part of $x$ and $\left\{ x\right\} =x-[x]$.

Problem : Let $n$ be a positive integer, for a real number $x$, let $[x]$ denote the largest integer not exceeding x and $\{x\} =x -[x]$ Then $$\int^{n+1}_1 \frac{(\{x\})^{[x]}}{[x]}dx$$ is equal to ...
0
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1answer
40 views

Find the indefinite integral $\int (a+x)^{2 \over 3} x^3 dx$

Find the indefinite integral $$\int (a+x)^{2 \over 3} x^3 dx$$ any hints?
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3answers
89 views

If $\int _{ 0 }^{ 1 }{ f(x) dx } \leq10$ then is it correct to say $\int _{ 0 }^{ 1 }{ {f(x) }^2dx } \leq100$? [closed]

If $\int _{ 0 }^{ 1 }{ f(x) dx } \leq10$ then is it correct to say $\int _{ 0 }^{ 1 }{ {f(x)}^2 dx } \leq100$ ? If not,why?Do provide counter examples if you can.Thanks.
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1answer
43 views

Integrate $ \int dr(dr+2r)\left(1-\frac{r+\frac{dr}{2}}{r_0}\right) $

How can I integrate the following? $$ \int dr(dr+2r)\left(1-\frac{r+\frac{dr}{2}}{r_0}\right) $$ Where $r_{0}$ is a constant and $r=[0, r_{0}]$
1
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0answers
48 views

Radial Green's function

I would like to solve an equation of the form $$ \bigg(\frac{d}{dr^2} + m^2 \bigg)f(r) = g(r), $$ for $f(r)$. Normally I would just find the Green's function $G(r,r')$, which is defined by $$ ...
0
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1answer
21 views

Inconsistent integration from mathematica and hand

$\int^{2\pi}_{0} \frac{x}{2}e^{imx} \, dx$ by hands give $\frac{\pi}{im}$ assume m is not zero. But in mathematica: I type: ...
14
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3answers
263 views

How to find $\int_0^1f(x)dx$ if $f(f(x))=1-x$?

A friend of mine gave this question to me : Suppose $f(x)$ is a real, non-constant, differentiable function satisfying the functional equation $f(f(x))=1-x$ ...
0
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1answer
53 views

Investigate absolute convergence of the integral $\int_0^\infty x^2\cos e^x\,dx$

I am studying absolute convergence of improper integral over $\left[0,+\infty\right)$ $$\int_0^\infty\!x^2\ \cos(e^x)\ dx$$ And I used the substitution $t=e^x$, I produce the improper integral ...
1
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1answer
58 views

Calculus (first year) challenging arcsin integration

I'm currently studying for a midterm, and this question has me stumped: Given that $\int_0^1(\arcsin x)^6,dx = k$, find the value of $\int_0^1(\arcsin x)^8,dx$ in terms of k. Simplify your final ...
0
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1answer
21 views

Value of this integral

$\int^{+\infty}_{0} e^{ikx-x} \, dx$? Making $ikx-x = x(ik-1)$, I calculate it as (infinity - 1) which is infinity? Is this true?
0
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1answer
26 views

Evaluating an integral without a parametrization

From what I understand one of the main benefits of differential forms over Riemann integrals is that you're supposed to be able to integrate differential forms without parametrizing your curve (or ...
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2answers
36 views

How to integrate $x^2/{\sqrt{16-x^2}}$ using trig substitution

I try first by substituting x for 4sin(u), but then i get stuck and im not sure what to do, thanks.
0
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0answers
40 views

Volume of a solid of revolution obtained from $y = 2x^2$ and a line

I want to find the volume of the solid of revolution obtained from rotating the region between: the curve formed by connecting the graph of $y=2x^2$ from $x = 0$ to $x = 2$ and the straight line ...
1
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1answer
83 views

Improper integral substitution hint

i try solve this improper integral $$\int_0^\infty x^p\sin x^q \ dx$$ I try to compare it with $\displaystyle \int_0^\infty\ \frac{1}{x^p}\ dx$ But I don't know what do when $x\rightarrow\ \infty$ in ...
0
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1answer
38 views

Integral of sine raised to a fractional power: $\int_\frac{-1}{2}^\frac{1}{2} \vert \sin(\pi f)\vert^{2d} df$

How would I go about evaluating the following integral? Let $d\in [-1/2,1/2]$. Its related to the spectral density function of a fractionally differenced processes. $$\int_\frac{-1}{2}^\frac{1}{2} ...
0
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0answers
8 views

how to obtain $\int_{r_1}^x \int_x^{r_1}D_m Dp^+ x u(s)\, dm(s)\, dp(y) = u_0(x) - D_p u_0(r_1)(p(x)-p(r_1))$?

in the book of Petr Mandl - Analytical treatment of one-dimensional Markov processes (pg 35) one reads: where (5) is just I can't quite find the same result as the author. This is how I ...
0
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1answer
28 views

Find the characteristic function of the distribution

$f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ and $f(x) = 0$ for $x <0$ Here is what I got: $$\phi_x(t)=\mathbb{E} [e^{itX}]$$ $$=\int_\mathbb{R} e^{itx} f_X (x) dx=\int_0^\infty (\cos x + ...
2
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1answer
49 views

Solving integrals looks like Fourier integrals(2)

I'm wondering how to obtain this integral: $$\int_0^\infty \frac{e^{-w^2 t}}{1+w^2} \cos(w x)\ dw$$ I've tried to set this integral a function of $t$ ($f(t)$) then I calculate the derivative of ...
2
votes
1answer
73 views

$\int_0^\infty\frac{K_0(x)K_0(\alpha x)}{K_0(\beta x)}\cos xy\phantom{.}dx$ Integral from 1926 electrotechnical paper

Erdelyi et.al "Table of integral transforms, vol. I" on p. 50 cites the following integral $$ \int_0^\infty\frac{K_0(x)K_0(\alpha x)}{K_0(\beta x)}\cos xy\phantom{.}dx, $$ but instead of printing the ...
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0answers
33 views

What is the inverse function?

$\lambda:\mathbb{R}^n\to\mathbb{R}$ smooth, $\lambda>0$. Let $$ F(s(t))=t_0+\int_{x_0}^{s(t)}\frac{1}{\lambda(x(w))}\, dw=t. $$ Hence $$ s(t)=F^{-1}(t), $$ isn't it? But what is $F^{-1}(t)$? I ...
1
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2answers
44 views

Evaluating the integral $\int (x^2-1)^{\frac{-3}{2}}dx$

I need to solve the differential equation: $$\displaystyle f'(x) = (x^2-1)^{-\frac{3}{2}}, f(2)=\frac{3-2\sqrt 3}{3}$$ Which basically amounts to solving the integral $\int (x^2-1)^{-\frac{3}{2}} ...
0
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1answer
18 views

Limit of an integral - Convergence

Assume $f : (0,1) \to \mathbb{R}$ be measurable and integrable in $(0,1)$. Find $\lim_{k \to \infty} \int_0^1 x^k f(x) dx $ Note that here, integrablity of $f$ in $(0,1)$ means $\int_0^1 f(x) ...
2
votes
1answer
41 views

$\int_0^T t^{2a}K_a(t)^2\;dt=\text{ ?}$ where $K_a$ is modified Bessel function of second kind?

Let $K_a$ be the modified Bessel function of second kind, with $a>0$ a real number. Is there a nice expression for $$\int_0^T t^{2a} K_a(t)^2\;dt,$$ where $T < \infty$? The expression for ...
1
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3answers
46 views

Calculate the volume of the region trapped by $z^2=x^2+y^2, z=2(x^2+y^2)$

Task Calculate the volume of the region trapped by $z^2=x^2+y^2, z=2(x^2+y^2)$ using a triple integral. I'm kind of lost on this one, here's my (probably wrong) attempt: Calculate the ...
3
votes
2answers
135 views

Motivation behind notation $\int_C P \, dx + Q \, dy$

I think the notation $$ \int_C P \, dx + Q \, dy $$ is a bit confusing. I understand fairly well the notation $\int_C \vec{F}\cdot d\vec{r}$ and I understand from my question here that they are the ...
1
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1answer
43 views

Is $\int_C P dx + Q dy = \int_C \vec{F}\cdot d\vec{r}$?

I am a bit confused about some notation. For a vector field $\vec{F}$, I understand the notation $$ \int_C \vec{F}\cdot d\vec{r}. $$ But I have also seen the notation $$ \int_C P dx + Q dy $$ If ...
2
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0answers
30 views

Double Integrals involving Bessel Function of zeroth order

I need some help solving the integrals below, $$ I_1 = \int_0^a \int_{0}^{R} k\left( {\partial \over \partial z} {\mathrm e^{-\mathrm i z \sqrt{a^2 - k^2}} \over \sqrt{a^2 - k^2} } \right) \cos(a ...
1
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2answers
45 views

Integration involving polynomial and exponential function

Question: Integrate the following function: $$\int \frac{(x^2+1)e^x}{(x+1)^2}\ \mathrm{d}x$$ I tried to simplify the function. However, that didn't get me anywhere. I'm aware of the identity: ...
3
votes
2answers
109 views

Derivative of a negative order?

Below, $\Delta$ means taking the derivative, $\frac{d}{dx}$. For $n\in\mathbb{Z}$, $n\geq 0$, we have $$\Delta^n\sin{x}=\sin{(x+n\tau/4)} \\ \Delta^n\cos{x}=\cos{(x+n\tau/4)}$$ I found that out while ...
8
votes
4answers
164 views

Prove: ${n\choose 0}-\frac{1}{3}{n\choose 1}+\frac{1}{5}{n\choose 2}-…(-1)^n\frac{1}{2n+1}{n\choose n}=\frac{n!2^n}{(2n+1)!!}$

Prove: $${n\choose 0}-\frac{1}{3}{n\choose 1}+\frac{1}{5}{n\choose 2}-...(-1)^n\frac{1}{2n+1}{n\choose n}=\frac{n!2^n}{(2n+1)!!}$$ $(2n+1)!!$ is a factorial of odd integers, ...
0
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0answers
17 views

Question about change of variables in multiple integral calculation

The question is to calculate this multiple integral: $$\iint\limits_{\Omega} {{\textrm{d}x \textrm{d} y}\over ...
0
votes
1answer
82 views

Calculate using integration by parts $\int \frac{x^2}{(x^2+1)^2}\,dx$

Calculate using integration by parts $$\int \dfrac{x^2}{(x^2+1)^2}$$ I'm looking through some working for this question and it gives $u=x/2, u'=1/2$ $v'=\dfrac{2x}{(x^2+1)^2}, ...
3
votes
2answers
59 views

Solving integrals looks like Fourier integral.

$$\int_0^\infty \frac{\sin w}w \, \cos xw \, dw$$ How can I solve this integral,I was thinking it may be solved by using Fourier transform, but It seems that it wouldn't work.besides I've tried many ...
1
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1answer
60 views

Fourier integration of $f(x)=\pi e^{-x}$

I've tried below Fourier integration and reached some answer.I would appreciate if anyone takes a look at this and enlighten me if something is wrong (or if it is right): $ f(x)=\begin{cases}\pi ...
0
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0answers
13 views

Can you find the integral quotient by vector projection?

Okay so last night I was in math class and we were learning about how to find the Integral Quotient for a tangent to a line in a coordinate system like in this picture: At one point the teacher ...
2
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1answer
79 views

How to integrate this with feynman method? [duplicate]

$$\int_0^{\infty} \frac{\sin^2(x)}{x^2(x^2+1)} dx$$ The integral is equals with $\frac{\pi}{4}+\frac{\pi}{4e^2}$, but i can't prove it.
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1answer
33 views

How can I find the value of…

Please help me finding the value of the following integral. If $U\subseteq \mathbb{C}$ is an open set and $z_0\in U$ and $r>0$ and $\{z:|z-z_0|\le r\}\subseteq U$ and $j\in \mathbb{Z}^+$ $$\large ...
1
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1answer
26 views

Latent Dirichlet Allocation Derivation

I am exploring different derivations for the the LDA and was a bit surprised about a step I found in the following paper : https://cxwangyi.files.wordpress.com/2012/01/llt.pdf My question is about ...
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0answers
20 views

Show that transformation $u=\frac{y}{x^3}, v=\frac{y}{x}$ determines a bijection between $(0, \infty)^2$ and $(0, \infty)^2$

I am working on a problem which states: Sketch the finite planar region $D$ which is bounded by the four curves $$y = x^3,\space\space y = 10x^3,\space\space y = x,\space\space y = 2x.$$ Show ...
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0answers
44 views

$L^p(\mathbb{R})$ separable.

I'm trying to prove $L^p(\mathbb{R})$, $p \in [1,\infty)$ is separable by showing the collection$$ S:= \{\sum_{i=1}^nr\chi_{(a_i,b_i)}\}_{(a_i,b_i,r) \in \mathbb{Q}^3}$$ is dense in $L^p$. So, since ...
1
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1answer
27 views

Splitting integrals in $L^p$

I know that for $f \in L^1$, and $\mu(S) > 0$, we have $$\int_X f \, d\mu = \int_{X \setminus S} f \, d\mu + \int_{S} f \, d\mu.$$ Is the same true in $L^p$? Or do we now get an inequality? ...
0
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1answer
63 views

Find volume of region lying above $z=0$, below $z=4-x^2-y^2$ and inside extruded disc $x^2+y^2=2^2$

I am working on the following homework problem: Find the volume of the region that lies above the plane $z=0$, below the surface $z=4-x^2-y^2$ and inside the extruded disc $x^2+y^2=2^2$. I think ...
0
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0answers
21 views

Integral parameter equation

How to solve the following equation, $$\int_{-i\infty}^{+i\infty} e^{st}\frac{\Sigma_{k=0}^{m}b_ks^k}{\Sigma_{k=0}^{n}a_ks^k} ds=0$$ to obtain $t$ where, $m\le n$ and $a_i$, $b_i$ are the given ...
0
votes
0answers
31 views

Proof that Absolute value of Riemann Integrable Function is Riemann Integrable using set of discontinuity points.

I'm aware that the statement has been proven true in another answer, but I was curious if this could be done using the requirement that a function $f$ is Riemann integrable iff it is bounded and its ...
2
votes
2answers
34 views

If $\int_1^\infty x^{-p} dx\,$ exists, then does $\int_1^\infty x^{-q} dx\,$ exist, where $q > p$?

If $\int_1^\infty x^{-p} dx\,$ exists, then does $\int_1^\infty x^{-q} dx\,$ exist, where $q > p$? My initial assumption is that the answer was true. Because if p is an arbitrary number 5, then ...
1
vote
1answer
30 views

If f is continuous and 0 < f(x) < g(x) on the interval [0, ∞) and $\int_0^\infty g(x) dx = M < \infty$ then $\int_0^\infty f(x) dx$ exists?

If f is continuous and $0 < f(x) < g(x)$ on the interval $[0, ∞)$ and $\int_0^\infty g(x) dx = M < \infty$ then $\int_0^\infty f(x) dx$ exists. True or False, and why? I'm not sure what the ...
3
votes
2answers
79 views

Dominated Convergence

Is there a function that dominates $$f_n(x) = \frac{1}{(1+\frac{x}{n})^nx^{\frac{1}{n}}}$$ on $(1,\infty)$ for all $n$? I need to apply DCT to get $e^{-x}$.Obviously MCT doesn't apply since ...
1
vote
0answers
25 views

Evaluating $\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz$ between balls in $\Bbb R^3$

I am working on the following problem: Evaluate: $$\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz,$$ where $R = \big\{(x, y, z) : 1 \leq x^2 + y^2 + z^2 \leq 2^2 \big\}$ is the ...
1
vote
0answers
29 views

Derivative of a definite integral from a parameter

Is there any way to calculate $a$ to satisfy $$\frac{\partial{F(a)}}{\partial{a}}=0,$$ where $$F(a)=\int_{-\infty}^{+\infty} f(x,a)dx$$ $f$ can be any function but we know that the above ...