Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

1
vote
1answer
27 views

Integral related to a geometry problem

In the question Geometry problem involving infinite number of circles I showed that the answer could be obtained by the sum $$\sum_{k=0}^\infty \int_{B_k} \frac{4}{|1 + (x + i y)|^4} \, dx \, dy,$$ ...
0
votes
0answers
18 views

Definite integral, substitution method

$\mathbf{Substitution~method:}$ Let $f\in C[a,b]$ , $\varphi\in C'[m,M]$ and $\varphi(m)=a,\varphi(M)=b$, then $$ \int_a^bf(x)dx=\int_m^Mf(\varphi(t))\varphi'(t)dt. $$ My question is isn't it enough ...
0
votes
4answers
38 views

General form for these types of integrals

I encountered this integral in physics-- $$2\int_{0}^{\infty} \dfrac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} dt$$ I know for certain that $a>0$, $b>0$. $a$ and $b$ are independent variables
0
votes
1answer
51 views

Alternative view on integration?

The question is about an alternative view on formulating or arriving at the concept of the integral (in case this is possible of course). Let's say we want to add a series of values $f(x_i)$ occuring ...
1
vote
1answer
21 views

Showing that $\tan(\pi z) = z$ has exactly three solutions in the strip $|\Re(z)| < 1$

We can't use Rouche's theorem here directly, so we have to apply the argument principle. If $f(z) = \tan(\pi z) - z$ , then $f'(z) = \pi \sec^2(\pi z) - 1$. Choose the rectangle $\Gamma$ with ...
0
votes
1answer
28 views

Explaining why $\int_{b-\frac{1}{n}}^{b-\frac{1}{2n}}\text{affin}=\text {area of triangle}$

Little earlier laid out an example of which can be found at: Explaining why $\Vert x_n\Vert _1=\frac{3}{4}$ I do not understand why: $$\int_{b-\frac{1}{n}}^{b-\frac{1}{2n}}\text{affin}=\text {area ...
1
vote
0answers
50 views

Evaluate $\frac{1}{\tau} \int\limits_0^\tau{{(e^{\alpha t}-1)e^{-\beta t}}}dt$

I have to evaluate $\frac{1}{\tau} \int\limits_0^\tau{{(e^{\alpha t}-1)e^{-\beta t}}}dt$ , where ατ=1. I did he following: But they have the following answer: Have I forgotten something? I did ...
0
votes
0answers
25 views

What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$ \int_{0}^\infty dx\, R(x), $$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
2
votes
1answer
27 views

Complex Green's Theorem

I want to integrate $\int_{\partial R} |e^{zt}|dz$ where $R\subseteq \mathbb{C}$ is a rectangle whose sides are parallel to the coordinate axes. I want to use a complex version of green's theorem, but ...
11
votes
1answer
114 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
1
vote
1answer
113 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
0
votes
0answers
24 views

Integration By Parts and convergence

I have an integral as $\int_0^\infty e^t w(t)f(t)dt$ where $f$ is a continuous PDF function and $w(t)\to 1$ at infinity. Is there any way to solve such an integral by parts? At the first glance ...
1
vote
1answer
57 views

To be integrable, a function must be bounded; yet $\int (1/x^3)\,dx = -1/(2x^2)$?

One condition of integrability is that the function is bounded across the interval. $1/x^3$ however has a pole at $x=0$ yet it's integral is defined over all real numbers. Doesn't this violate the ...
5
votes
1answer
102 views

Is there a theory of integration in elementary terms for definite integrals?

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For ...
9
votes
3answers
459 views

Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...
0
votes
2answers
35 views

Find $\iiint_E sin^3 x+\tan y+ 6\hspace{1mm} dV$, where $V$ is region inside $x^2+y^2+z^2 = 1$

I guess that the integral of $\sin^3 x+\tan x$ part is zero, because i have seen many problems like these where the integral is over a symmetrical region and the functions are odd. But I want ...
5
votes
1answer
108 views

A strange answer for $\int _{-1}^1 logx dx$

I typed the integral of $\int _{-1}^1 logx dx$ on wolfram alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening. Thanks.
4
votes
0answers
54 views

Integral's Closed-form expression in terms of hypergeometric function

I want to solve the following integral: $$I = 2\left[\int_{0}^{1}\dfrac{y^m}{(1 - ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y+\int_{0}^{1}\dfrac{y^m}{(1 + ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y\right]$$ ...
4
votes
3answers
84 views

Calculating $\int_{0}^{0.5} \frac{\ln(1+x)}x$ with at least two decimal places precision.

I'm preparing for a test and i've spent quite some time on this. What I already did was to use the taylor expansion for $\ln(1+x)$ to finally get the sum: $$ \sum_{n=1}^\infty \frac{(-1)^{n-1}}{{2^n} ...
0
votes
0answers
12 views

Definite integral - Finding an equivalent form

I have the following definite integral $ \int_{0}^{L} {\psi(t) }_{1 \times 5}{A(s)}_{5 \times 5}(\psi(t) _{1 \times 5})^{T} {B(s)}_{5 \times 5} ds \tag 1 $ Given data All dimensions are ...
5
votes
0answers
144 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
4
votes
5answers
189 views

Integral of $1/[(1+x^2)\sqrt{1+x^2}]$

I try to get back on track with the integration. I would like to solve $$ \int_0^1 \frac{dx}{(1+x^2)\sqrt{1+x^2}}.$$ There are my way to try to solve it (that I don't find the right solution) and an ...
1
vote
3answers
76 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
1
vote
0answers
33 views

When is limit of integral equal to integral of limit?

Under what conditions does the following equation holds: $\lim_{h \to 0} \int_0^1 f(z+ht)\,dt$=$ \int_0^1 \lim_{h \to 0} f(z+ht)\,dt$
3
votes
1answer
44 views

Solving linear non-homogeneous integral equation

Is it possible to solve equations of the kind: $$x(t) = \int \limits_t^T c_1 x(s) ds +c_2 $$ with $c_1$ and $c_2$ being some constants and if I know $x(0)$? Or do I need more assumptions on $x$?
3
votes
1answer
98 views

How to evaluate the integral $\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$

I would like to evaluate the following integral: $$\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$$ I get the Integral by Maple and it gives the Lommel function. After that, I will search ...
4
votes
2answers
84 views

Integral of $R(R^2+y^2)^{-3/2}$ with respect to $y$

$$\int_0^\infty \frac{R}{\sqrt{R^2+y^2}\left(R^2+y^2\right)}dy$$ The indefinite integral seems to be $$\frac{-R}{\sqrt{R^2+y^2}}+C$$ $R$ is a constant
5
votes
0answers
72 views

Dealing with an integral: can we go any farther?

I meet an integral, but it is beyond my ability. $$ {\rm I}\left(a\right) = \int_{a}^{1}{\arcsin\left(\,\sqrt{\,{1 - x^{2} \over 1 - a^{2}}\,}\,\right) \over x + 1}\,{\rm d}x, 0\le a <1. $$ I can ...
2
votes
0answers
70 views

asymptotic expansion of the integral for large tau

How can I proceed to resolve this integral? $$ c_1\int_{-\infty}^{\infty}{\frac{\cos\left(x\tau\right)}{\left(1 + c_{2}\,x\right)^{\alpha}}}\, \,{\rm d}x $$ where $c_1, c_2$ are positive constants, ...
4
votes
2answers
120 views

Definite integral $\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$

Hi guys I have the following definite integral to solve: $$\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$$ is it possible to obtain an analytic expression? And if not why? ...
1
vote
1answer
61 views

Integration question.

The question is as follows For any real number $x$, let $\lfloor{x}\rfloor$ denote the greatest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval ...
0
votes
1answer
41 views

Integral of two function multiplied [closed]

Please let me know what / how does the result of below integral results would differ: $$(f \star g)(\tau) = \int_{-\infty}^\infty f(t)\star g(t+\tau)\,dt,$$ $$(f + g)(\tau) = \int_{-\infty}^\infty ...
0
votes
0answers
37 views

Definite integral with dx=d

I'm currently reading a paper (Baye & Morgan & Scholten, 2006. "Information, Search, and Price Dispersion") where the following integral is computed (p.14) : $\int^{r}_{\underline{p}}p^2 ...
2
votes
2answers
210 views

An Improper Integral

I need help with this integral: $\Large {\int_0^\infty \frac{dx}{x\sqrt{1+x}}} $ What I did: Substitute $\sqrt {1+x} = t$. Then the integral turns into $ \int_1^\infty 2dt/(t^2-1) $. Now I replaced ...
3
votes
1answer
50 views

Why is the value of $\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|\,dx$ independent of integer parameter $n$?

I am not able to find an easy solution for the following formula $$\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|dx=4+\frac{4}{3}\pi\sqrt{3}.$$ Please help me prove it. Why it does not depend on the (positive) ...
0
votes
0answers
41 views

Integrating by parts with exponential and power-law functions

I have a question about integrating by parts for $$\int_{L}^{U}\left[x^{a} \cdot e^{-bx}\right]\,dx$$ for positive reals $L,U$ with $L<U$ ($L, U \in [0, +\infty) $). I'm interested in cases with ...
1
vote
2answers
91 views

Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle

Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y ...
0
votes
1answer
126 views

Finding the value of $\int_{0}^{1} \frac{\sin^2 x}{x^2}dx$

I would like to find the exact value of $$\int_{0}^{1} \frac{\sin^2 x}{x^2}dx.$$ First of all we know that it exists and must be $\hspace{0.1cm}$$\leq1$$\hspace{0.1cm}$ because$\hspace{0.1cm}$ ...
0
votes
0answers
46 views

Integral of $\exp(-x\,f(x))$

What is the evaluation of the integral of the following form or is there any alternative form for it? $$\int e^{-x \, f(x)} dx \tag 1$$
12
votes
2answers
338 views
+100

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
9
votes
4answers
215 views

How to evaluate a certain definite integral: $\int_{0}^{\infty}\frac{\log(x)}{e^{x}+1}dx$

How can I show that: $$\int_{0}^{\infty}\frac{\log(x)}{e^{x}+1}dx=-\frac{\log^{2}(2)}{2}$$ EDIT: This is equivalent to showing that $\eta'(1)=-\ln2\gamma-\dfrac{\ln^2(2)}{2}$.
8
votes
1answer
51 views

Changing the order of integration without sketching?

When changing the order of double integrals, I have always relied on sketching the region. I have recently come across this example on MSE by @FelixMartin which seems to avoid visual-based reasoning, ...
4
votes
2answers
159 views

Integral of the Von karman equation

What is the result of this integral, and how can I proceed: $$ \int_{-\infty}^{\infty}{c_{1} \over\left(1 + c_{2}\,x^{2}\right)^{5/6}}\, \cos\left(x\tau\right)\,{\rm d}x\,,\qquad c_{1}, ...
1
vote
2answers
70 views

An intergral with variable upper limit

Let $$\psi \left(x \right)=\int_{0}^{x}\frac{\ln(1-t)}{t}dt,x\in (0,1).$$ Show $$\forall x\in (0,1), \psi\left(x \right)=?$$ I return the old variable $t$ by the substitution $s=ln(1-t)$,and then ...
-1
votes
1answer
61 views

Some confusing and tough (for me) integrations [closed]

Can anyone please help me with these integrations : $\int_0^3$ $| x+1 |$ $dx$ $\int$ $(|x-2|+|x-1|+|x|+|x+1|+|x+2|)$ $dx$ $\int$ $|x|dx$ $\int$ $(e^{|x|}$ + $\ln x)$ $dx$ $\int_0^\pi$ ...
5
votes
0answers
66 views

Existence of the integral $\int_{0}^{\pi}\frac{\sin (x+\sqrt x)}{\cos(x-\sqrt x)} dx$

Can anyone help me in evaluating the following integral : $\int_{0}^{\pi}\frac{\sin (x+\sqrt x)}{\cos(x-\sqrt x)} dx$. I tried doing this by substitution but didn't work. Also by parts looks ...
6
votes
1answer
71 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
1
vote
1answer
62 views

Evaluation of $\int_0^\infty \frac{(x^2+y^2)^{-s/2}}{e^{2\pi y}-1}\cos(s \arctan(y/x))dy$

$$\mbox{Does the integral}\quad \int_{0}^{\infty}{\left(x^{2} + y^{2}\right)^{-s/2} \over {\rm e}^{2\pi y} - 1}\, \cos\left(s\arctan\left(y \over x\right)\right)\,{\rm d}y\quad \mbox{converge or ...
6
votes
2answers
136 views

Find the length of the curve $x^{2k}+y^{2k} =1$

I want to find an expression for length and find the limit $k\rightarrow \infty$ The answer is obviously 8, if we look at the graphs.
0
votes
1answer
34 views

How to calculate the cumulative density of a bump function

How would I go about calculating the integral $$\int_{1/4}^x \exp\left(\frac{-1}{1-16(t-\frac{1}{2})^2}\right)dt,$$ where we assume $\frac{1}{4} \leq x \leq \frac{3}{4}$? Thanks!