All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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69 views

Stone's Theorem Integral

Given a finite Borel measure $\mu$ and a Hilbert space $\mathcal{H}$. Consider a strongly unitary group $U:\mathbb{R}\to\mathcal{B}(\mathcal{H})$. Introduce for simple vector-valued functions: ...
11
votes
3answers
86 views

An equivalent for $\sum_{n=0}^{\infty} e^{-x\sqrt{n}}$ as $x$ tends to $0^+$

I would like to obtain an equivalent form for $$ f(x)=\sum_{n=0}^{\infty} e^{-x\sqrt{n}} $$ as $x \rightarrow 0^+$. I tried without success to "remove" the $\sqrt{\cdot}$ in the summand by summing ...
5
votes
3answers
511 views

Square integrable function that doesn't go to zero?

I'm reading through some elementary quantum mechanics textbooks and a few authors mention that "there exist pathological functions that are square-integrable but do not go to zero at infinity." ...
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1answer
22 views

Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition and a nonexample: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample The Riemann integral ...
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0answers
19 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
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0answers
17 views

Problem with this question on solid of revolution

Calculate the volume of a revolution solid obtained by rotation around the x-axis, the region bounded by the graph of $y=e^x$, $-1\le x \le1$ and the x-axis. Thanks in advance, and sorry about my ...
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2answers
80 views

Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$

How do I integrate this? $$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$ I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of ...
5
votes
1answer
86 views

Banach Spaces: Uniform Integral vs. Riemann Integral

Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Predefine a simple integral: ...
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0answers
21 views

Piecewise vs Continuous Integration

I have the following data: Daily spend on marketing Daily gain of fans because of that spend on marketing ('billed' fans) The 'organic' daily number of fans for the same period above (ie free ...
3
votes
3answers
53 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
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0answers
10 views

Simple indefinite integral of a vector function

I am having trouble with this simple integration. I am not sure of the process or steps to follow to solve this type of problem: If $\mathbf{V}(t)$ is a vector function of $t$, find the indefinite ...
5
votes
2answers
113 views

Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$

Here is an interesting series I played with, namely $$\sum_{n=1}^{\infty} \frac{\displaystyle\psi\left(\frac{n+1}{2}\right)}{\displaystyle \binom{2n}{n}} \approx -0.245969181104090562617616399148$$ ...
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0answers
14 views

When substituting in integration, do you have to change the limits of integration so long as you keep it consistent?

I have this integral: In order to solve for it, I have to substitute: t=tan(theta) dt=(sec(theta))^2 d(theta) When substituting that, I know I have to change the limits of integration within ...
7
votes
3answers
115 views

Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$

In my course, I have to prove formula below $$I=\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$$ for $a,b,c>0.$ I know that ...
0
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2answers
91 views

Evaluating $\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} dy$

I was trying to find arc-length of $x = \sqrt{y}-7y$ So basically right now I am stuck with this $$\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} \,\mathrm dy$$ $$\int^{4}_{1} ...
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1answer
49 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
3
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1answer
77 views

Generalized Riemann Integral: Nonexample?

Definition Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. (In fact, a Hausdorff TVS should be sufficient.) Consider functions $F:\Omega\to E$. Define the generalized ...
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5answers
96 views

Show that $\displaystyle\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$

$$\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$$ My Intuition telling me there might be an $\arctan$ coming up, but I don't know how to do this ...
10
votes
5answers
199 views

Ways to prove $\displaystyle \int_0^\pi dx \dfrac{\sin^2(n x)}{\sin^2 x} = n\pi$

In how many ways can we prove the following theorem? $$I(n):= \int_0^\pi dx \frac{\sin^2(n x)}{\sin^2 x} = n\pi$$ Here $n$ is a nonnegative integer. The proof I found is by considering ...
2
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2answers
68 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
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0answers
43 views

Integration of a polynomial

I am facing a problem in finding the integral $$\int\frac{r^2}{-C r^3 + r^2 -2 M r +Q^2}\,dr$$ Here M, Q, and C are parameteres (to be fixed later). Could anybody Please help me in finding it? I ...
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1answer
25 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
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1answer
22 views

Uniform convergence and integrability

If $(f_n)_{n \in \Bbb N}$ converges to $f$ uniformly and each $f_n$ integrable would it imply $f$ is integrable and $$\lim_{n \to \infty}\int f_n = \int f$$ In case each $f_n$ is nonnegative ...
3
votes
3answers
53 views

Problems with this integral $ \int \sqrt{1 + {1 \over t^2} + {2 \over t}} dt$

$$ \int \sqrt{1 + {1 \over t^2} + {2 \over t}}\,\mathrm dt$$ I tried making substitution, using $ u=1 + \dfrac{1}{ t^2} + \dfrac{2 }{ t} $, then , $dt=\dfrac{du}{-2\left({1 \over t^3 }+ {1 \over ...
13
votes
6answers
459 views

Proving $\displaystyle\int_0^1\frac{\ln(x)}{x^2-1}\,\mathrm dx=\frac{\pi^2}{8}$

How can I prove that? $$\int_0^1\frac{\ln(x)}{x^2-1}\,\mathrm dx=\frac{\pi^2}{8}$$ I know that $$\int_0^1\frac{\ln(x)}{x^2-1}\,\mathrm ...
17
votes
3answers
831 views

Evaluate the integral $\int_0^{\infty} \left(\frac{\log x \arctan x}{x}\right)^2 \ dx$

Some rumours point out that the integral you see might be evaluated in a straightforward way. But rumours are sometimes just rumours. Could you confirm/refute it? $$ ...
5
votes
2answers
50 views

Integration $\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R$.

Give a compact form for the solution of integral: $$\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R,k\in\mathbb N$$ any suggestions please?
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votes
2answers
23 views

Arc length of a curve which already has an integral sign

This one here was tricky because the function already has an integral sign. My guess is that I need to evaluate the indegral where $x=4$ so that i get $y=f(t)$ and after that apply the Arc Length ...
1
vote
0answers
36 views

Antiderivative of $|x − 2| + |x − 3|$ [on hold]

Find the most general antiderivatives of the following function. $$|x − 2| + |x − 3|$$ I started with showing that the antiderivative for $|u|$ is $\dfrac{u|u|}2$. How to proceed then?
3
votes
0answers
316 views

The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to ...
0
votes
1answer
51 views

The limit of $ m\int_{a}^{1/m} \frac{dx}{x}=0 $ and $ m\int_{a}^{\infty} \frac{dx}{x^{1+m}}=0$ as $m\to0$

Given $ a >0 $ is it correct that $$ \lim_{m\to 0}m\int_{a}^{1/m} \frac{dx}{x}=0 $$ by the properties of the logarithm function? Or on the other hand, $$\lim_{m\to 0} m\int_{a}^{\infty} ...
0
votes
1answer
25 views

Evaluating line integrals in the plane

Evaluate the integral $$\int xy\,dx+(x+y)\,dy$$ along the curve $y=x^2$ from $(-1, 1)$ to $(2, 4)$. I tried finding $dx$ and $dy$ and substituting that into the original integral, along with $y=x^2$. ...
1
vote
1answer
281 views

Integrating using the residue theorem

How would I calculate the following integral? $$\int_{-\infty}^\infty \frac{1}{(x^2 + 1)(x^4+4)^2} dx$$ Part (a) says define Laurent's theorem for the Laurent series expansion and give the definition ...
2
votes
1answer
24 views

Partial Derivative of a nonexistant variable?

I am wondering how I would find the partial derivative of $z = g(r, \theta) = \theta$ with respect to both $r$ and $\theta$. I realize that if you take the partial in respect to $\theta$, it is 1. I'm ...
2
votes
0answers
23 views

The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
8
votes
3answers
212 views

Indefinite Integral of Reciprocal of Trigonometric Functions

How to evaluate following integral $$\int \frac{\mathrm dx}{\sin^4x+\cos^4x\:+\sin^2(x) \cos^2(x)}$$ Can you please also give me the steps of solving it?
2
votes
0answers
44 views

McShane vs. Henstock-Kurzweil: Lebesgue Integrability

Put in words, is it right to say that the difference of the McShane integral to the Henstock-Kurzweil integral is that the tags are not required to lie within $x_i\leq t_i\leq x_{i+1}$? If so, is ...
3
votes
1answer
90 views

Gauge Integral: well defined?

Given a compact space $\Omega$ and a Banach space $E$ Consider functions $f:\Omega\to E$. Regard neighborhood gauges: ...
0
votes
1answer
30 views

Verify Green's Theorem for region bounded by the lines $x=2$, $y=0$, $y=2x$

Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, $y=2x$ and the functions $f(x,y)=(2x^2)y$, $g(x,y)=2x^3$. I have been trying this question for far too long and I can't ...
-3
votes
1answer
50 views

Taking the Derivative of $F(x)=\int_0^x f(t)\,dt$ [on hold]

Let $F(x)=\int_0^x f(t)\,dt$ What is the derivative of $F(x)$? I desperately need guidance!
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0answers
52 views

Magnus series expansion

In the theory of the Magnus series expansion, it can be found that $$ \Omega(t) = \int_0^t A(\tau)d\tau - \frac{1}{2}\int_0^t \left[ \int_0^\tau A(\sigma)d\sigma, A(\tau) \right]d\tau + ...
0
votes
1answer
34 views

Lebesgue integration in one variable

I have studying the conditions for the existence of the Lebesgue integral. Generally, to show that existence of the integral of a function on an unbounded interval, one can integrate and take ...
2
votes
1answer
26 views

Inequality involving Holders Inequalities

Suppose $f\in L^p(\mathbb{R})\cap L^\infty(\mathbb{R})$ for some $p>2$, show that $||f||_{p}\leq ||f||_2^{2/p}||f||_{\infty}^{1-2/p}$ I tried to write $|f|^p=|f|^{\frac{p}{2}}|f|^{\frac{p}{2}}$ ...
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votes
0answers
39 views

What are the integration of these inverse trigonometric function? [on hold]

Integrate the following: Please Help me, I don't where to start. I used several methods to solve this like completing the squares.. $\int\frac{u^4+4}{u^4+9}du$ $\int\frac{\sin(x)(\cos ...
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votes
1answer
25 views

Prove that the antiderivative of an integrable function is both bounded and integrable

Let $f: [a,b] \to \mathbb{R}$ be a bounded function which is also integrable. Define $F: [a,b] \to \mathbb{R}$ by $$F(x)=\int_{a}^xf(t)\ dt$$ To prove that $F(x)$ is also bounded and integrable I ...
4
votes
1answer
66 views

Integral of cosine over a quadratic

I need help with the following integral: $$ \int_{-\pi}^{\pi}{\cos\left(\, ax\,\right) \over 1-bx^{2}}\,{\rm d}x $$ The constants $a$ and $b$ are both real and positive. Any help will be ...
1
vote
1answer
343 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
-1
votes
0answers
8 views

A cylinder with base radius of 9 𝑐𝑚 and height 16 𝑐𝑚 is cut by the plane [on hold]

A cylinder with base radius of 9 𝑐𝑚 and height 16 𝑐𝑚 is cut by the plane 𝑥 + 𝑦 + 𝑧 = 9 and 𝑧 = 0. Using triple integral, find the volume of the cylindrical section bounded by the planes.
1
vote
0answers
16 views

Question on area under and between curves and volume of a solid by revolution

I have recently begun learning about finding the area under the curve by definite integrals. But I am still a little unsure of the concepts. When you integrate for a certain range of the graph , does ...
10
votes
2answers
171 views

Elegant proof of $\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$?

Let $a, b > 0$ satisfy $a^2-4b^2 \geq 0$. Then: $$\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$$ One way to calculate this is by computing the residues at the ...