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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37 views

Integral of a square root quadratic with negative leading coefficient

I have this homework problems: $$\int \dfrac{dx}{\sqrt{-x^2 + 3x - 4}}$$ What i did was take out $\sqrt{-1}$ from denominator, and complete square. The result I get was : $$\dfrac{\ln\left|x+ \sqrt{...
-1
votes
1answer
78 views

How to find the function such that $\int_0^1f(x)\ \mathrm dx=e^{-4n^{2}{\pi}}$ [on hold]

Find $f(x)$ where: $$ \int_{0}^{1}f(x,n)\ \mathrm dx=e^{-4n^{2}{\pi}} $$ Is it possible that question contains infinitely many answers? How to solve this ? Please provide me a hint.
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1answer
64 views

prove that integral of laplacian squared equals to integral of sum of integral of partial deritatives squared

Let f be a $C^3$ function, $f: \mathbb{R}^n \to \mathbb{R}$, f is compactly supported prove that: $$\int_{\mathbb{R}^n} (\Delta f) ^2 = \sum_{i,j = 1}^n \int_{\mathbb{R}^n} (\frac{\partial^2f}{ \...
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1answer
36 views

evaluate the integral where C is the…

I just want to check the answer.Thank you! Evalulate $$\int_C (y+\sin x)dx + (z^2+\cos y)dy +x^3dz$$ where $C$ is the curve given parametrically by $$r(t) = \langle \sin t, \cos t, \sin 2t \rangle$$, ...
-2
votes
0answers
20 views

A complex no. as the limit of integration [on hold]

I have come across an integral equation, integral (f(x)dx) limits from 0 to i=sqrt(-1) and f(x) is a real function. Now, I know the value of integral(f(x)dx) limits from 0 to real number a. Also, f(x) ...
5
votes
3answers
79 views

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $?

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $ ? I tried substituting $x=1/t$ but that's making it more complicated.Any suggestions?
2
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3answers
59 views

Is it correct to interpret the “dx” in the standard notation for integrals as the Lebesgue measure?

Ok so I am in my Calc I class for the summer and we are just beginning to talk about integrals. I know a little bit about measure theory and the Lebesgue integral and why is it more general than the ...
2
votes
1answer
38 views

Integral of Bessel Functions Multiplying “polynomials”

How can I compute the following integral: $$\int_{0}^{1} (1 - x^{2})^{\nu - \mu - 1} x^{\mu + 1} J_{\mu}(\alpha_{\nu}x) dx$$ where $\nu > \mu \geq 1$ and $J_{\nu }(\alpha_{\nu}) = 0$. The $J_{\nu}$ ...
1
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1answer
103 views
+100

On a solution to a triple integral.

I want to calculate the function $f(x,y,z) = z$ on the set $B = \{ (x,y,z) \in R^3 | z \ge \sqrt{7x^2 + 3y^2}, 2x + z \le 3 \}$ I tried to solve it without cylindrical substitutions. the solution is ...
1
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1answer
26 views

Difficulty setting up a solid of revolution integral

I am trying to find the volume of the region between the parabola $y^2=8x$ and the line $x=2$ revolved around the $y$ axis. My intuition is that taking an infinitesimal horizontal slice of the ...
3
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1answer
37 views

Finding the limits of integration in triple integrals without using the figure.

As an example I was trying to integrate the function $f(x,y,z) = x$ on the set $B = \{(x,y,z) \in R^3 \mid 4y^2 + z^2 \le 16, x \le \sqrt{4y^2 + z^2}, x+y \ge -1 \}$. I notice that by swapping to ...
5
votes
3answers
153 views

How do I prove $\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}{\cos(a-b)}$?

How do I prove these? $$\int_{-\infty}^{\infty}{\sin(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\sin(a-b)}\tag1$$ $$\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\...
2
votes
2answers
99 views

Integrating $\cos (x+\sin (x))$

I tried to solve $$\int\cos(x+\sin(x))\,dx$$ but it seems to be way out of my league (tried u-substitution with $u=x+\sin(x)$ and couldn't find an answer). Also, no one on the Internet seems to have ...
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0answers
22 views

Volume by 3 Planes

I'm new to Integration Problem. For now i'm pretty messed up with this Volume Problem. Can anyone please revise it! Calculate the Volume of interception of three planes which is given by equation $z^...
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1answer
34 views

Lebesgue integral of $\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}$ on an infinite cylinder

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell, and let $\boldsymbol{r}\in\mathbb{R}^3$ be any point of the space. I intuitively suppose that the Lebesgue integral $$\...
4
votes
0answers
92 views

$\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ [on hold]

Find the value of: $\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ I do not really know where to start, so please forigve me for not showing my attempt. Wolfram alpha gives $2....
2
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1answer
26 views

Finding arc length of the curve $6xy=x^4+3$ from $x=1$ to $x=2$

Looking at this as a graph of a function of $y$ is more convenient $$ y=\frac{x^4+3}{6x}\Rightarrow \frac{dy}{dx}=\frac{x^3-1}{2x^2}\Rightarrow \left( \frac{dy}{dx} \right)^2=\frac{x^6-2x^3+1}{4x^4} ...
1
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1answer
19 views

Periodic Function with Integral

Problem: $f(x)$ is a continuous function, and it is periodic with period $T$. For any $a<b$, prove that $$\lim_{n\to\infty}\int_a^bf(nx)dx=\frac{b-a}{T}\int_0^Tf(x)dx$$ I tried substituting ...
1
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0answers
20 views

Volume of a geodesic ball in a Riemannian manifold with $K<0$.

Let M be a simple connected Riemannian Manifold wih $K_M < 0$. Prove that the volume of any geodesic ball of M is strictly greater than $\frac{Vol(S^{n-1})r^n}{n}$, where $n = dim(M)$ and $r$ is ...
0
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1answer
54 views

More complicated uSubstitution

I have absolutely no idea what to do here other than use uSubstitution. $$\int{{1}\over{4x^2 + 9}}\mathrm dx$$ I also tried looking at the output of an integral calculator but to no avail. I noticed ...
2
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1answer
44 views

Need help with an integral involving the Dirac delta function

I'm having trouble evaluating this integral, which involves the Dirac delta function: $$ \int\limits_{0}^\infty \frac{\cos(\pi x)}{x} \delta \left[ (x^2-1)(x-2) \right] \mathrm{d}x $$ I think I ...
3
votes
3answers
115 views

Does $\int_0^{1/2} \frac{1}{x\ln x}dx$ converge?

I tried this: $$ \begin{align*} \ln x &= t \\ \frac{1}{x} dx &= dt \\ \lim_{x \to 0^+} \ln x &= -\infty \end{align*} $$ So now we have $$ \int_{-\infty}^{\ln(1/2)} \frac1t dt $$ which ...
0
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0answers
22 views

line integral problem, stuck in the curve

Problem: A force field is given in polar coordinates by the equation $$F(r,\theta)= (-4\sin (\theta), 4\sin (\theta)).$$ Compute the work done in moving a particle from the point $(1,0)$ to the ...
0
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1answer
21 views

Upper sum is strictly subadditive

Can you give me an example demonstrating that the upper sum is strictly subadditive (i.e. (the upper sum of $f$) + (the upper sum of $g$) is strictly bigger than the upper sum of $(f+g)$)?
4
votes
1answer
67 views

Proving that a function is integrable

Given that $f:[0, \infty] \to \mathbb{R}$ is decreasing with $\displaystyle\lim_{x \rightarrow \infty} f(x)=0$, prove that $$I=\int_{0}^{1}\frac{\cos(\frac{1}{x})f(\frac{1}{x})}{x^2}dx$$ converges. ...
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1answer
103 views

How to evaluate $\int\left\{\log\left(\log\left(x\right)\right) + {1 \over \left[\log\left(x\right)\right]^{2}}\right\}\,\mathrm{d}x$?

$$ \mbox{How to evaluate ?}\quad \int\left\{\log\left(\log\left(x\right)\right) + {1 \over \left[\log\left(x\right)\right]^{2}}\right\}\,\mathrm{d}x $$ Hints or suggestions please.
0
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0answers
56 views

Evaluating an integral

I have problem to evaluate the following integral: $$I=\int_{0}^{2\pi} \frac{\sin^2\phi\, d\phi}{a+b \cos{\phi}}$$ In fact I calculate it using complex integration methods, by changing the variable : $...
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0answers
6 views

$e^{-x^2}M_{k,m}(x^2) \in L_1$ space?

Let $M_{k,m}(z)$ be m-Whittaker Function defined in enter link description here. How can show that $e^{-x^2}M_{k,m}(x^2),\, m,k > 0$ is belongs to $L_1\left(\mathbb{R}\right)$ or not? Thank you ...
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2answers
40 views

Showing a function is integrable [on hold]

Let $\xi, \zeta\in\mathbb{R}^m$. How might one try to show that $f:\mathbb{R}^m\times\mathbb{R}^m\rightarrow \mathbb{R}$, defined by $\displaystyle\frac{1}{(1+\left|\xi - \zeta\right|)^{k}}$ is or is ...
13
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2answers
244 views

Daunting series of integrals: $\sum_{n=2}^\infty\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log(\frac{1-\sin x}{1+\sin x})dx$

My coleague showed me the following integral yesterday \begin{equation} I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
0
votes
3answers
84 views

Differential Equations (Coffee)

This is a long post so bear with me until I get to the part where I am stuck on! :) Question: The author of a popular detective novel drinks black coffee to help him stay awake while writing. ...
0
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1answer
21 views

Integral Inequality with Monotonic Function

Problem: For continuous, either both increasing or both decreasing functions $f, g$ on $[a, b]$, suppose that $p(x)$ is continuous and positive. Prove that $$\int_a^bp(x)f(x)dx \int_a^bp(x)g(x)...
3
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0answers
16 views

Defining derivatives and integrals for hyperoperations > 2

Derivatives and Integrals are continuous generalizations of the Forward Difference and Summation additive operators respectively. We can do the same with multiplication and get multiplicative calculus ...
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0answers
26 views

Fubini's theorem (interchange of sum and integrals) in case of multivariable function

Can the Fubini's theorem in case of single variable sequence of functions be readily extended to multivariable sequence of functions?, i.e, Is it true to say $$\iiint_V\sum_{n=0}^\infty f_n(u,v,w) \,...
1
vote
0answers
23 views

Kline Calculus intuitive approach Chapter 3 problem 12

The problem is as follows : Water drops flow out from a small opening at the rate of one drop per second and fall vertically with an acceleration of 32 ft/sec^2. Determine the distance between two ...
1
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1answer
33 views

Evaluate the integral $\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt$

I have a question in solving the integral $$\int_{-\infty}^{\infty}{\left| 2t\cdot\text{sinc}^2(2t)\right|^2}\,dt.$$ I know that you can use Parseval's Theorem to prove that $\int_{-\infty}^{\infty}\...
5
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3answers
262 views

Integration by parts or substitution?

$$\int_{}^{}x e^x \mathrm dx$$ One of my friends said substitution , but I can't seem to get it to work. Otherwise I also tried integration by parts but I'm not getting the same answer as wolfram. ...
0
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1answer
54 views

Compute definite integral by hand [on hold]

How can I compute $$\int_0^1 \frac{x^3t}{(x^2+t^2)^2} \, \mathrm{dt}$$ by hand?
3
votes
2answers
46 views

Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
9
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3answers
215 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{\big(\frac{x}{1-x}\big)}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)...
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2answers
52 views

Given $f(x,y)$ is a continuous function, Do these integrals equal? [on hold]

Given range $\{ 0 \le x \le 1, 0 \le y \le 1\}$ Do these integrals equal? $\int_0^1(\int_0^y f(x,y)dx)dy = \int_0^1(\int_0^x f(x,y)dy)dx$ Well, the answer is no. It seems like the triangulars are ...
2
votes
2answers
67 views

Closed form for an integral with log and power

Let $n \in \mathbb{N}$. We know that: $$\int_0^1 x^n \log(1-x) \, {\rm d}x = - \frac{\mathcal{H}_{n+1}}{n+1}$$ Now, let $m , n \in \mathbb{N}$. What can we say about the integral $$\int_0^1 x^n \...
0
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0answers
21 views

numerical integration asymptotic relation

Let $Q\subset R^n$ be a convex subset and $f\in C^2(Q)\;$ We set $x_s:=\int_Q xdx$,$\;\;\;Vol(Q):=\int_Q 1dx$ and $diam(Q)=sup||x-y||_2$ Prove the following asymptotic relationship: $...
6
votes
1answer
106 views

Evaluate $\int \frac {\sin(x)}{x^2 + 4x + 5}dx$

Question: Evaluate $$ \int \frac{\sin(x)}{x^2 + 4x + 5} dx=\int \frac {\sin(x)}{(x + 2)^2 + 1}dx $$ By using the change of variable $y = x + 2$ we have that $dy = dx$ then $$I = \int \frac{\...
3
votes
4answers
87 views

Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
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votes
5answers
88 views

Evaluation of $\int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$

Evaluation of $\displaystyle \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$ $\bf{My\; Try::}$ Let $$I = \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx\;,$$ Put $x+1=t\; $ Then $dx = dt$ and changing limits, we get ...
1
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0answers
22 views

Matlab double integration result does not match with my self calculate

Here is a double integration, self learning $$ P=\int_{-w}^{w}\int_{l}^{\frac{y_h(x_b+w)}{x_h}+l}\frac{1}{2}\operatorname{erfc}\left[\frac{\log{\frac{z_h(y_b-l)}{y_h}}-\mu}{\sigma\sqrt2}\right]\space ...
-2
votes
1answer
40 views

What are the solutions to integration problems below? [on hold]

$$\int (a^2 - y^2)y dy $$ $$\int \frac{e^{\sqrt{x}} + 1}{\sqrt{x}} \ dx$$ $$\int \frac{x^3}{\sqrt{1-2x^2}} \ dx $$
-1
votes
0answers
34 views

show the if f(x)'>0 the integral exist, help please. [on hold]

I need to show that if f(x) is Superior monochrome in [a,b] the integral for f(x) in [a,b] exist, from here I know that if f(x) is Superior monochrome for all x2,x1∈[a,b], (x2>x1), f(x2) ≥ f(x1) I ...