5
votes
3answers
51 views

Why is $\int^{\infty}_{0}{(1-\cos x)\over{x^{2}}}dx = \frac\pi{2}$

I have been having trouble understanding Fourier series and analysis in one of my classes. This is one problem from the text and we have to show that this is true. I have done other problems related ...
0
votes
2answers
26 views

Calculus power series

Hi could anyone help me to solve this. express the function $\int_x ^0 (\sin(t^2)\cdot \cos(t^2))$ as a power series. Because there is two trigo identies I do not know how to combine them to form a ...
2
votes
2answers
23 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
1
vote
0answers
33 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
8
votes
1answer
202 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
0
votes
1answer
22 views

How to solve this definite integral from fourier series?

I am stuck with this integral: $$\int\limits_0^{2L} \sin \left( \frac{m \pi x}{L} \right) \sin \left( \frac{n \pi x}{L} \right)\, dx.$$ How to solve this integral? In general, can you refer me to ...
3
votes
2answers
62 views

Integral question with square roots

I can't find the solution for this integral. I don't know why I didn't succeed in solving it. I will be really glad if someone can help me find the answer to the following: ...
2
votes
0answers
79 views

integrate a difficult function

I can't solve it. please help! I tried everything. Integration by parts - doesn't work. but maybe I didnt do it right. I tried to substitute , but I'm stuck. $$\int \frac{x}{\cos x}\sin(\tan ...
1
vote
0answers
54 views

Evaluate $\int_a^s\frac{(t-s)^n}{t(t-z)^{n+1}}dt.$

Evaluate: $$\int_a^s \frac{(t-s)^n}{t(t-z)^{n+1}}dt,$$ where $n>0$ is an integer, $0<a<s$. I'm not sure what values $z$ should take yet, but any help with the integral would be helpful. I ...
4
votes
0answers
89 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
2
votes
2answers
50 views

Problem calculating line integral

I have $\gamma=[0,1]\to\mathbb{R}^3$ defined by $\gamma(t)=(\cos(2\pi t), \sin (2\pi t), t^2-t)\;\forall t\in[0,1]$ and I'm asked to calculate ...
1
vote
0answers
63 views

How to solve this integral ($\int _{\frac{\pi }{6}}^{\frac{\pi }{4}}\sqrt{1-\tan ^2\left(x\right)}dx$)

$$\int _{\pi/6}^{\pi/4}\sqrt{1-\tan ^2\left(x\right)}dx$$ Hey, can you help me to solve this integral please? Thanks.
1
vote
0answers
41 views

Computing the values of $ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{2n} $

Define the following integral as $$ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{n}\,,\quad V_\alpha(n):=\int_0^\pi x^{\alpha-1}\cos(x)^{n} $$ where $n \in\mathbb{N}$. Now in the base case ...
3
votes
0answers
34 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
3
votes
2answers
67 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
0
votes
1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
2
votes
1answer
31 views

Re-interpreting double integral as a Type II Region $\mathrm{d}y\,\mathrm{d}x$ vs $\mathrm{d}x\,\mathrm{d}y$

I have the following Double Integral:$\iint_Dx\cos y\space\mathrm{d}A$ where $a$ is bounded by $x=1,y=0,y=x^2$. Interpreting this region as a Type one region, it is easy to conclude $R=\{(x,y)\mid ...
1
vote
1answer
37 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
1
vote
0answers
28 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
2
votes
1answer
65 views
+50

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
2
votes
0answers
41 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
6
votes
1answer
130 views

How to prove $\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}$?

How do you prove that $$\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}\;\;\;\left(=7 \zeta(3)\right)~?$$ p.s. Mathematica gives a pretty good ...
8
votes
0answers
202 views
+400

$\int_0^1 [ \frac{1}{x(x-1)} (2\mathrm{Li}_2(\frac{1-\sqrt{1-x}}{2})-\log(\frac{1+\sqrt{1-x}}{2})^2 ) -\frac{\zeta(2)-2\log^2 2}{x-1} ]dx$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
0
votes
1answer
47 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
2
votes
2answers
59 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
108 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
51 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 ...
6
votes
2answers
101 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
32 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 $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
10
votes
2answers
163 views

Calculate $\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$

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 started. ...
0
votes
1answer
49 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
231 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
67 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 ...
1
vote
1answer
48 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!
9
votes
0answers
197 views
+400

$I=\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
0
votes
3answers
30 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
38 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 - ...
7
votes
3answers
677 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
34 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
55 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
54 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
57 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
45 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 ...
1
vote
0answers
45 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
31 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
32 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 $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...