0
votes
0answers
10 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) e^{xcos[\pi (t-1)/2]} dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) \end{equation} as $x\rightarrow0$. ...
2
votes
2answers
48 views

Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$

Hi I am trying to show $$ \int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx=\frac{\pi}{32\alpha^{3/2}},\quad \Re(\sqrt \alpha)> 0. $$ I am looking for a solution to this NOT using contour integration, but ...
7
votes
3answers
89 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
2
votes
1answer
42 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
5
votes
2answers
91 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
1
vote
0answers
28 views

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Hi I'm trying to show that $$ I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}. $$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
4
votes
0answers
54 views

Calculate the following Integral (Please Help)

Hi, I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me ...
0
votes
1answer
45 views

How to integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$?

How do I integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$, where $C_1$ is an arbitrary constant? Is this integral really complex (hard to integrate)? EDIT: This comes from DE: $dy/dx = ...
0
votes
1answer
26 views

How do I integrate this in terms of error function

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think ...
3
votes
3answers
221 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi= \int_0^1 \frac{x^4\,\left(1-x\right)^4}{1+x^2}$$
5
votes
1answer
57 views

How to evaluate $\int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt$?

$$ \int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt $$ I tried doing it by parts and looking for differentials but I just keep getting back to the original expression. I can't think of a clever ...
0
votes
1answer
42 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
2
votes
0answers
41 views

Integral $\int_0^{\pi/3}\log\bigg( \frac{1+2\cos\theta}{2}+\sqrt{\left( \frac{1+2\cos\theta}{2} \right)^2-1}\ \bigg)d\theta.$

Hi I am trying to calculate this integral I given by $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
0
votes
3answers
29 views

How to get from $3\int_{-1}^0 (x^3-x) dx \,\,\,- \,\,\, 3\int_0^1 (x^3-x) dx$ to $6\int_{-1}^0(x^3-x)dx$?

Homework problem: Set up the definite integral that gives the area of the region. Two functions are given: $y_1 = 3(x^3-x)$ $y2 = 0$ The graph of $y1$ runs from x=-1 to x=1. I've gotten this ...
1
vote
1answer
28 views

Bessel's integral, how to actually evaluate?

I am just about to study Bessel functions and I have recently seen one of its integral representations given by: $$ J_ \alpha (x) = \frac{1}{\pi} \int_0 ^ \pi \cos(\alpha \tau - x\sin\tau) d\tau - ...
6
votes
3answers
661 views

How do I solve this definite integral?

$$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$ I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows: $\sin^{4}x + \cos^{4}x = (\sin^{2}x + ...
0
votes
2answers
40 views

Integral of $e^x ln(e^{2x} - 4)$

Find the integral from ln4 to ln6 of $$e^x \ln(e^{2x} - 4)$$ I factored $$\ln(e^{2x} - 4)$$ to get $$\ln((e^{x} - 2)(e^{x} + 2))$$ Then I separated this to get: $$e^x\ln(e^{x} - 2) + e^x\ln(e^{x} + ...
2
votes
1answer
32 views

A bounded integral

I want to show that there exists $K\in\mathbb{R}^+$ such that $$\left|\int_{1}^x \sin(t+t^7)dt \right|<K$$ for all $x\ge 1$. Intuitively, I'm quite sure it is true, but I can't find a formal proof. ...
1
vote
1answer
36 views

Integral Involving $(\arcsin x)^3$

Find the integral of $$\int_{0}^{1}8(\arcsin x)^3 dx$$ Okay, so I have substituted $u = \arcsin x$. I multiplied the integral by $$(1-x^2)^{1/2}/(1-x^2)^{1/2}$$ And then I said that $$\cos u = ...
1
vote
1answer
27 views

Disappearing negative signs when evaluating a sinh^-1 integral

$$\int_{-2}^{6} \frac{1}{\sqrt{1+(-x)^2}} \, dx$$ When performing this integral on paper, I get $$\sinh^{-1}(6) - \sinh^{-1}(-2) $$ But when I type it on wolframalpha, I get the unintuitive answer ...
1
vote
3answers
54 views

Find the integral of $\frac{x^5+x^2+4x+\sin(x)}{64+x^6} dx$ from $-2$ to $2$

Find $$\int\limits_{-2}^{2}\frac{x^5+x^2+4x+\sin(x)}{64+x^6}dx$$ I understand this questions is trying to make the point about the integrals of symmetric functions. So I separated all of these to ...
-1
votes
3answers
53 views

Solving an Integral With Square Root in the Denominator [duplicate]

Solve for k: $$\int_6^{16}\frac{1}{\sqrt{(x^3 + 7x^2 + 8x -16)}}dx = \frac{\pi}{k}$$ I have factored the denominator to obtain $x^3 + 7x^2 + 8x - 16 = (x-1)(x+4)^2$ I think that since the answer ...
0
votes
0answers
55 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
1
vote
0answers
43 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
0
votes
0answers
41 views

LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)d\theta$

I am trying to integrate the Log Sine Integral: $$ Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^nd\theta $$ where n is a non-negative integer. This problem is strongly ...
0
votes
0answers
29 views

LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

This integral is known as the moments for the generalized log-sine integrals. The notation I am using is similar to Lewin and what he used in the 1950's-1980's. $$ ...
1
vote
0answers
31 views

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
2
votes
1answer
109 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
2
votes
1answer
31 views

Volume vs. Surface Area Integrals

In order to find the volume of a sphere radiud $R$, one way is to slice it up into a stack of thin, concentric disks, perpendicular to the $z$-axis. a disk at any point $z$ will have radius ...
2
votes
1answer
74 views

Integrate $I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx$

Hi I am trying to evaluate the integral $$I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx.$$ Note we can write the integrand as $$ x^2 \ln\big(\frac{e^x-e^{-x}}{2}\big) ...
3
votes
2answers
109 views

some problems with evaluating $\int_0^{\pi}\ln(\sin x+\sqrt{1+\sin^2x})dx$

solve $$\int_0^{\pi}\ln(\sin x+\sqrt{1+\sin^2x})dx$$ when i start to solve it I think with $\sin x=u$ but I faced a $\color{#00f}{problem}$ its $\sin \pi = \sin 0 = 0$ so the integral is equal to ...
3
votes
1answer
106 views

How to evaluate the integrals: $\int_0^{\pi/2}(\cos^2 \frac{x}{2}+x\cos x)e^{\sin x}dx$

$$I_1=\int_0^{\pi/4}\frac{\sin x}{x\cos^2 x}dx$$ With this integral, I can't solve. I think setting $x=\frac{\pi}{4}-t$ is ok. But it seeems to be wrong. $$I_3=\int_0^{\pi/2}(\cos^2 ...
0
votes
0answers
37 views

How to solve integral equations like this?

sorry for such a non-specific question and lack of research effort, but I'm new to integral equations and don't know where to start. How does one go about solving equations of the form ...
3
votes
3answers
80 views

Proof of definite integral $\int_0^{\pi/2} \frac{\sin(2n+1)x}{\sin x}dx=\frac\pi2$ using induction

Prove by induction or otherwise that $$\int_0^{\pi/2} \frac{\sin(2n+1)x}{\sin x}dx=\frac\pi2$$ for every integer $n\ge0$. How to prove the above question? Can it be proved without using induction?
2
votes
1answer
109 views

Integrate $\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta$

Hello I am trying to integrate $$ I=\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta $$ which is similar to Integral...$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d ...
0
votes
1answer
39 views

Are the two integrals equivalent?

Consider $x \in \mathbb{R}$, $A\subseteq \mathbb{R}$, $f(x)$ continuous in $\mathbb{R}$ and the integral $$ g(A):=\int_{x \in A}^{} f(x) dx $$ Is $g(A)$ equal to the integral $$ ...
6
votes
4answers
262 views

Does the improper integral exist?

I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit $$ \lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ doesn't exist. I thought about ...
2
votes
2answers
122 views

Cool Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
0
votes
1answer
50 views

Help with finding the definite integral of $e^{\frac{2x-x^2}{2}}$?

I have this integral that I am trying to evaluate by hand, but I am encountering some difficulties. According to Wolfram Alpha, the answer seems to be: However, I do not understand how they got ...
2
votes
0answers
56 views

Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx,\quad \alpha,n \geq 1. $$ Thanks. This integral is old and has a 4th order pole. I am also looking for literature ...
1
vote
0answers
28 views

Integral$\int_{-\infty}^\infty x^{2n} e^{-\beta (x^2+\cos x+\alpha x)}dx$

Hi I am trying to integrate $$ \int_{-\infty}^\infty\int_{-\infty}^\infty (xy)^{2n}\exp\left({-\beta(x^2+y^2+\cos x+\alpha x+iy)}\right)dxdy \quad \alpha,\beta,n >0. $$ These integrals can be ...
1
vote
1answer
36 views

Integral $\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$

$$ I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n. $$ Evaluate I for $n \to \infty$ and z real. We can consider $z\geq 0$ due to the symmetry of $\mathcal{F}$ given by $$ ...
1
vote
0answers
45 views

Integrate $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old and has a triple pole. I am also looking for ...
1
vote
1answer
36 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
3
votes
1answer
124 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
0
votes
3answers
54 views

How can I integrate $\sqrt{\sin^2{2t}}$?

How can I integrate $\int_0^\frac{\pi}{2}\sqrt{\sin^2{2t}}~\mathrm{d}t$?
2
votes
3answers
75 views

Integrate $ x\cdot e^{i\omega x - x^2}$ from $0$ to $\infty$

Can anybody help me in solving the following integral: $$\int_0^\infty x\cdot e^{i \omega x-x^2}\,dx\quad?$$ Any help/hints will be highly appreciated. Thanks
0
votes
1answer
31 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
0
votes
0answers
40 views

Solving the indefinite integral $\int\frac{dx}{(e^x + e^{-x})^2}$ by conjugates

Given to solve: $$\int\frac{dx}{(e^x + e^{-x})^2}$$ For one method - the first step to take would be to multiply by conjugate $$\int\frac{(e^x)^2}{(e^x)^2}\frac{dx}{(e^x + e^{-x})^2}$$ My ...
3
votes
1answer
44 views

Area of part of parametric function

I need to get area of function: $x= 2\sqrt{2}\cos ^3 t$ and $y= 4\sqrt{2}{\sin ^3 t}$, but only the part when $x\geq1$. How can I do that? I know that area of full function would be $$S= \int_a^b ...