Questions about the evaluation of specific definite integrals.

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-1
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1answer
28 views

Explanation of the passage from $\int_{N'}^N dN/N$ to $\ln N-\ln N'$

While going through my text I got stuck in the derivation given in the picture. ($\Omega$ is a constant) I don't know how to get the second step from the first step, also I don't know why ln is ...
0
votes
2answers
26 views

Power series solution to integral equation

Hi guys i'm reading a paper in which the authors have two coupled integral equation for the function $f(x)$ and $g(x)$, in order to solve this problem they employ a power series expansion of these ...
0
votes
0answers
34 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
-1
votes
2answers
62 views

Integral of inverse of square root of a quadratic

I haven't taken a course on calculus so far so I don't know what to do. The integral may be wrong. Please tell me which part of it is wrong. $$ q∫_{+a}^{-a}\lim_{c \to g}\frac 1{(b^2+c^2)^{3⁄2}} dc $$ ...
2
votes
4answers
56 views

I'm wondering when the integral $\int_{-\infty}^\infty \frac{dx}{(1+x^2)^\alpha}$ converges.

I'm wondering when the integral $$ \int_{-\infty}^\infty \frac{dx}{(1+x^2)^\alpha} $$ converges for the real number $\alpha$.
6
votes
3answers
58 views

How to find the integral $\int_0^{70 \pi} |\cos^{2}x\sin x|\,dx$?

I need help with this problem: $$\int_0^{70 \pi} \left|\cos^{2}\!\left(x\right)\sin\!\left(x\right)\right| dx$$ My friend says it's 140/3 but I don't see how.
3
votes
3answers
61 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
4
votes
2answers
72 views

improper integral containing $\sqrt{\cos x-\dfrac{1}{\sqrt 2}}$ in the denominator

How do i find the value of this integral-- $$I=\displaystyle\int_{0}^{\pi/4} \frac{\sec^2 x \ dx}{\sqrt {\cos x-\dfrac{1}{\sqrt 2}}}$$ I came across this integral too in physics.
2
votes
0answers
49 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) \end{equation} ...
2
votes
1answer
127 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}} {4 \over \,\left\vert\,1 + \left(\,x + ...
0
votes
0answers
22 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
59 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
1
vote
0answers
70 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
29 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
55 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 ...
14
votes
1answer
168 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
117 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
26 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
61 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
465 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
37 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 ...
1
vote
2answers
70 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
6
votes
2answers
139 views

A strange answer for $\int _{-1}^1 \log x\; dx$

I typed $\int _{-1}^1 \log x\; dx$ on Wolfram Alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening?
4
votes
0answers
56 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
86 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 ...
6
votes
0answers
147 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
78 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
101 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
85 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
74 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
73 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
121 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
42 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 ...
4
votes
4answers
61 views

Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int$ $ \left \lfloor{x}\right \rfloor $ $dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left ...
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
127 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? $$\int e^{-x \, f(x)} dx \tag 1$$