Questions about the evaluation of specific definite integrals.

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6
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2answers
35 views

Definite integral of product of functions

I know it's not correct to write: $$\int_{a}^{b}f(x)g(x) dx = \int_{a}^{b}f(x)dx\int_{a}^{b}g(x)dx$$ This result seems obvious, but I can't think of a way to prove that $\int_{a}^{b}f(x)g(x) dx$ ...
0
votes
1answer
18 views

What happens when the interval of an integral changes from infinity to a constant number?

There exist a calculation about electromagnetic mass: $$m_\mathrm{em} = \int {1\over 2}E^2 \, dV = \int\limits_{r_e}^\infty \frac{1}{2} \left( {q\over 4\pi r^2} \right)^2 4\pi r^2 \, dr = {q^2 \over ...
2
votes
0answers
34 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
9
votes
1answer
85 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
1
vote
1answer
38 views

Changing argument into complex in the integral of Bessel multiplied by cosine

I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind. I found the integral ...
0
votes
0answers
46 views

Double Integration Working Help

Help I dont know how to approach this question, I have the answer but dont know how to write a detailed working process of obtaining it. It is supposed to find the surface area of a cone that is $z = ...
2
votes
2answers
60 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
0
votes
1answer
17 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
9
votes
0answers
85 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
1
vote
1answer
15 views

Limit of Cosine and Sine Fourier Transforms

If I define the cosine and sine Fourier transform as (skipping constant prefactors $(2\pi)^{0.5}$): $$\mathcal{F}_C\{f(x)\}=\int_0^{\infty}\,f(x)\,\cos(\omega x)\,dx$$ and ...
0
votes
0answers
27 views

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (t-s)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
0
votes
0answers
37 views

MAple 17 won't evaluate my integral

I type this into maple and it won't evaluate it: $$ \int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy $$ I've also tried $$ evalf(\int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy)$$ It just returns for both cases $$ ...
0
votes
2answers
43 views

Definite integral from -1 to 0 [on hold]

How would I evaluate this definite integral $$ \int_{-1}^{0}\tan x dx- \int_{-1}^{0}\sin^2 x dx $$ All i need to know is what to do when an integral is on an interval of -1 to 0. I could do this ...
3
votes
0answers
47 views

Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
1
vote
1answer
32 views

Definite integral including natural log, cosine, and hyperbolic sine

Here is an integral question I have, I am solving some other problems like this but I am stumped on this one: $$\int_0^{\pi+1}\frac {\ln(\cos(x+1))}{\sinh(x^2)}dx$$ I used some methods such as ...
7
votes
2answers
158 views

Definite integral, $\frac 1{\ln(x)}$

What is $$\int_0^{\pi^{2}}\frac 1{\ln(x)}dx$$ I tried using complex residues and some identities, but no luck. Any suggestions?
3
votes
2answers
75 views

Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...
3
votes
2answers
38 views

Evaluating a limit involving a definite integral

I want to prove the following limit evaluates to $0$ without using any techniques that involve complex numbers. I already solved it using residues and it's pretty straight forward, but it feels rather ...
4
votes
0answers
64 views

How to find the value of this integral?

This integral to the value \begin{align} \int_0^1\frac{\ln^2(1+x)\ln^2 x}{1-x}\ dx=&\ ...
1
vote
3answers
94 views

Trigonometric Substitution in $\int _0^{\pi/2}{\frac{ x\cos x}{ 1+\sin^2 x} dx }$

Evaluate $$ \int _{ 0 }^{ \pi /2 }{ \frac { x\cos { (x) } }{ 1+\sin ^{ 2 }{ x } } \ \mathrm{d}x } $$ $$$$ The solution was suggested like this:$$$$ SOLUTION: First of all its, quite obvious to have ...
4
votes
0answers
36 views

$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $ Integration

I am struggling to find the integration of the expression below, $$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $$ where $a$ and $b$ are arbitrary constant ...
1
vote
1answer
30 views

Asymptotic of a complex integral

Consider the following integral $$f(x):=\int_x^{+\infty}re^{-(r+ir^2)}dr$$ I want to understand the asymptotic behavior of $f(x)$ as $x\rightarrow +\infty$ Thank you for any suggestion.
1
vote
1answer
59 views

$\int_{0}^{\frac{\pi}{4}} e^{\sec x} \frac{\sin( x + \frac{\pi}{4})}{(1 - \sin x) \cos x}\, dx$?

How do I find the value of $$ \int_{0}^{\frac{\pi}{4}} e^{\sec x} \dfrac{\sin\Big( x + \dfrac{\pi}{4}\Big)}{(1 - \sin x) \cos x} \;\mathrm{d}x $$
3
votes
2answers
93 views

I was Stumped by this Challenging Integration/Limit Problem

Can someone show me how I do the following: \begin{align} I_n & = \int_0^1 \sqrt{ \frac 1 x + n^2 x^{2n} } \ \mathrm dx \\[8pt] & \lim_{n\rightarrow \infty } I_n = \text{ ?} \end{align}
2
votes
1answer
53 views

Help with an Inverse Trigonometry Integral 2

Evaluate $$\int^{1/{\sqrt{3}}}_{-1/{\sqrt{3}}} \frac{x^4}{1-x^4}\cos^{-1}\frac{2x}{1+x^2} \mathrm{d}x\\= \frac{\pi}{a}\ln(b+\sqrt{c}) +\frac{\pi^{d}}{e} - \frac{\pi}{\sqrt{f}}$$ Then Find ...
2
votes
1answer
75 views

Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } = \dfrac { \pi { \ln}^{ A }(B) }{ C } -\dfrac { { \pi }^{ D } }{ E } $$ $$$$ This was one solution, but it went completely ...
0
votes
1answer
41 views

Computing $\int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d} dx$

I want to find the integral of \begin{align*} \int_{0}^1 \frac{\left( (1-x )a +x b\right)^2}{(1-x)c +x d}dx \end{align*} for any $a,b$ and $c>0$ and $d>0$. Using Wolfram-Alpha I found that ...
1
vote
2answers
115 views

Cannot understand an Integral

$$\displaystyle \int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{ \sec x+\csc x } } =\frac { \sqrt { a } -b }{ 2 } +\frac { \sqrt { c } }{ 2 } \log(\sqrt { d } +\sqrt { e } -\sqrt { f } -g)$$ I had to solve ...
0
votes
2answers
32 views

Limit integral w/ sine function

Here's a random problem I thought of: $$\lim_{\alpha\to 0}\int_0^{\infty}\sin(\alpha x) \;\mathrm{d}x$$ What I'm trying to create is a sine function with an "infinite period", meaning instead of the ...
5
votes
1answer
42 views

Fourier transform in three dimensions getting out of hand

I have the following integral I wish to compute, it transforms a quantum position wave function into momentum space: $$\phi(\mathbf p)=\int\frac{\mathrm d^3r}{(2\pi\hbar)^{3/2}}e^{-i\mathbf{p\cdot ...
0
votes
0answers
27 views

Double Integral to evaluate volume over region [on hold]

I'm not sure how to write the integral needed for this problem: Find the volume of the solid bounded by the graphs of the equations: $$x^2 + z^2 = 1 $$ $$y^2 + z^2 =1 $$ And the first octant. I ...
0
votes
1answer
40 views

Calculate: $I=\int_2^5 \frac{\ln(x^2+1)}{x}dx$ [on hold]

Calculate: $I=\int_2^5 \frac{\ln(x^2+1)}{x}dx$ I used wolframalpha.com and get the result: $I=\dfrac{Li_2(-4)-Li_2(-25)}{2}$ Who can find this result's representation with primary function?
6
votes
3answers
135 views

A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$

I came across an interesting problem that I do not know how to solve: Find $x>0$ such that $$\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}.$$ Could you ...
3
votes
0answers
66 views
+50

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
0
votes
1answer
64 views

Prove the integral $\int_0^\infty \frac{x^n e^{-\mu x}}{x+\beta} dx$

I am trying to prove this integral here: Where Ei is the exponential integral. Unfortunately I don't have the right result yet, but I have other result that is not for this case, but I think it can ...
0
votes
0answers
32 views

Need proof of integration of sine parametrized functions [duplicate]

Yesterday, i encountered an integral formula (actually it's a generalization, i think). This : $$\int_0^\pi x f(\sin x)\,dx = \frac{\pi}{2} \int_0^\pi f(\sin x)\,dx$$ For simple functions like ...
0
votes
1answer
15 views

Infinite Integral of a Bessel Function

I need to calculate the following integral $$ \int_0^{\infty}xdxJ_n(kx) $$ Integrating it by parts and using the normalization of Bessel functions, I find it (somewhat heuristically) to equal the ...
5
votes
2answers
101 views

How to compute $\int_{-1}^{1} e^{-1/(1-x^2)}dx$?

As in the title, I would like to compute the integral: \begin{equation} \int_{-1}^{1}e^{-1/(1-x^2)}dx \end{equation} My hunch tells me that I should try to transform it to the correspoding ...
1
vote
2answers
20 views

Convert equation to Riemann sum to definite integral

How do you convert this equation to Riemann sum then to definite integral $$ \lim_{n \to \infty } \frac{1}{n} \Bigg(\sqrt{\frac{1}{n}} + \sqrt{\frac{2}{n}} + \sqrt{\frac{3}{n}} + ... + ...
3
votes
1answer
20 views

Problem about sequence of Riemann integrable function

a) Suppose that $g_n\ge0$ is a sequence of integrable function and $\int_{a}^{b} g_n(x) dx$ converges to $0$. If $f$ is an integrable function on $[a,b]$, then $\int_{a}^{b} f(x)g_n(x) dx$ converges ...
0
votes
2answers
27 views

Two approaches to the Volume of a Drilled Sphere (Self-Answered, open to other answers)

Find the ratio of the volume of a sphere of Radius $R$ with a hole through its vertical axis in the shape of a coaxial cylinder with radius $r$, to the original volume of the the sphere with raidus ...
0
votes
0answers
12 views

Interpretation of infinitesimal measure in Lebesgue integration

I have a little trouble understanding the notation of the infinitesimal measure in Lebesgue integration. For example, let's assume I want to compute an volume integral of a function $f: D \rightarrow ...
3
votes
1answer
45 views

Closed forms for definite integrals involving error functions

I have been working for a while with these kinds of integrals $$\int_0^\infty dx\,\text{erfc}\left(c +i x\right)\exp \left(-\frac{1}{2}d^2x^2+i cx\right)$$ $$\int_\Lambda^\infty ...
0
votes
1answer
32 views

Express each of the following integrals in terms of the gamma or beta functions and simplify when possible

$$\begin{align} \\ &= \int_{0}^{1} \ ( {(1/x) - 1} ) ^ {1/4} dx \\ &= \end{align}$$ I try to solve but not sure if correct or not and need help in these also ,I try in first one ...
1
vote
0answers
31 views

Can summation $\sum_{n=[-N\ldots N]}n e^{-\frac{(y-cn)^2}{2}}$ be lower bouned by integration $\int_{-N}^Nx e^{-\frac{(y-cx)^2}{2}} dx $

I was wondering if the following summation \begin{align*} \sum_{n=[-N \ldots N]}n e^{-\frac{(y-cn)^2}{2}} \end{align*} can be lower bounded by integral $$ \int_{-N}^Nx e^{-\frac{(y-cx)^2}{2}} \, dx ...
-3
votes
0answers
32 views

Use power series to approximate the definite integral $\int_0^{0.5} \frac{1}{1+u^5}\,du$ [closed]

Use power series to approximate the definite integral with an error less than $0.000005$: $$\int_0^{0.5} \frac{1}{1+u^5}\,du$$ Can you please walk me through it/ explain the concept? I'm having a ...
0
votes
0answers
32 views

Definite integral of absolute value is zero iff . . .

Given $f$ is bounded on nondegenerate interval $[a,b]$. I need to prove : If $f$ is continuous on $[a,b]$, then $\int_{a}^{b} |f|=0$ if and only if $f=0$ for all $x$ in $[a,b]$. The left direction ...
1
vote
2answers
52 views

An example of discontinuous integrable function

"Let $f(x)=1$ if $x=1,1/2,1/3,1/4,...$ and $f(x)=0$ elsewhere. Prove that $f$ is integrable on $[0,1]$. What is the value of that integral?" I'm guessing the value to be $0$, intuitively. I know the ...
2
votes
1answer
50 views

Calculate $\int_{-T}^T {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau$.

Let $\lambda$ and $\nu$ be real numbers. Then, it has \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= ...
2
votes
2answers
26 views

Integrate a sum (geometric series) round |z| = 1

This is a question from a text book (Saff and Snider, Complex analysis for mathmatics science and engeneering, page 203). Let $$ f(z) = \sum_{k=0}^\infty (k^3/3^k)z^k $$ Evaluate $$ ...