Questions about the evaluation of specific definite integrals.

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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 = ...
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3answers
19 views

Integral $\int_{0}^{\pi}\sqrt{1+4\sin^{2}(x/2)-4\sin(x/2)}\mathrm{d}x$

Here's how I solved it. \begin{eqnarray*} & & \int_{0}^{\pi}\sqrt{1+\left(4\sin^{2}\left(\frac{x}{2}\right)\right)-\left(4\sin\left(\frac{x}{2}\right)\right)}\mathrm{d}x\\ & = & ...
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1answer
23 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 ...
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1answer
35 views

integration $\int_a^b$$f$$(x)$$dx$ when $f(a)=f(b)$

How to integrate $\int_a^b$$f$$(x)$$dx$ when $f(a)=f(b)$ Can something like $\int_0^kf(x)dx=2\int_0^{k\over2}f(x)dx$ for $f(x)=f(k-x)$ be somehow used?
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3answers
213 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi= \int_0^1 \frac{x^4\,\left(1-x\right)^4}{1+x^2}$$
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1answer
50 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 ...
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2answers
44 views

Definite Integral $\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$

$$\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$$ This is easy to evaluate with complex analysis but is there an elementary way (substitution, partial fractions, integration ...
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1answer
41 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!
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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 $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
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26 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 ...
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2answers
30 views

Showing $f(z_0) = \frac{1}{2 \pi} \int_0^{2 \pi} f(z_0+Re^{i\theta}) \ d\theta$

Suppose that $f$ is analytic on and inside the circle $|z-z_0|=R$. Show carefully that $f(z_0)$ is equal to the average of $f$ on ${|z-z_0|=R}$, i.e. show that $$f(z_0)={1 \over {2 \pi}} \int_0^{2 ...
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1answer
14 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 - ...
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3answers
642 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 + ...
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1answer
34 views

Proof of integral equality

Let $f^{(n)}(x)$ be the $n$-th derivative of $f(x) = \cos(x)$. Prove that : $$ \int_0^{2\pi} f^{(n)}(x) \,\, dx = \int_0^{2\pi} f^{(n)}(kx) \,\, dx, $$ where $n$, $k$ are natural numbers equal or ...
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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} + ...
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1answer
31 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. ...
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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 = ...
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1answer
38 views

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
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2answers
41 views

Area under the curve $f(x) = \sin x$

Find the area under the curve $f(x) = \sin x$ on the interval $[0, \pi]$ if $\sin x \ge 0$ My handbook give this as $$\int_0^\pi \sin x \space dx = (\cos \pi) - (\cos 0) = (-1) - (-1) = 2$$ what ...
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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 ...
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3answers
53 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 ...
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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 ...
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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 $$ ...
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$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

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 ...
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4answers
211 views

Prove that these two curves have the same length

My midterms are approaching, and I was going through some of our past Calculus midterms when I stumbled upon this question from 1996: Show that these two curves, $$(\Gamma) : \frac ...
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2answers
39 views

An integral related to the power and algebraic function

I meet with a complex integral problem, given as follws: $$\int_0^\infty {\frac{{{x^p}}}{{{{\left( {x + a} \right)}^q}}} \cdot {{\left( {\frac{{x + b}}{{x + c}}} \right)}^n}dx,where{\text{ }}q > p ...
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40 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 ...
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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. $$ ...
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29 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 $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
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107 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) ...
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1answer
36 views

Definite Integral

Solve the following : $$\int_0^1 x^5 \sqrt{1 - x^2}dx $$ Any ideas ?I haven't been able to make any progress on this exercise , it's driving me insane , since there's probably no big deal to the ...
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1answer
21 views

Definite integral in spherical coordinates

I want to compute the volume enclosed by the sphere $x^2+y^2+z^2\le4$ with $0\le z\le 1.$ If I use the rings method I get: $$\pi \int_0^1 \left( \sqrt{4-z^2}\right)^2dz=\frac{11}{3}\pi $$ If I set ...
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1answer
30 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 ...
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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) ...
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108 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 ...
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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 ...
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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 ...
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2answers
76 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?
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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 ...
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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 $$ ...
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4answers
261 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 ...
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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 ...
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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 ...
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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 ...
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1answer
38 views

Integration of trigonometric functions times a simple rational function using residues

In the course of my research I have found a few integrals that I would like to have closed-form answers to: $$\int_{c- i \infty}^{c+ i \infty} \frac{1}{z-1} \frac{8 \pi^4 \cot{ \big( \frac{\pi}{6} z ...
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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 ...
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1answer
35 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 $$ ...
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44 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 ...
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3answers
57 views

How do I evaluate this limit with an integral function?

Could anybody give me some pointers on how do I evaluate this limit: $$\lim_{t \to +\infty} \frac{\int_0^t x^9e^{-x^2}\,dx}{\int_0^t x^7e^{-x^2}\,dx}$$ I'm not asking for the complete solution, just ...
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1answer
35 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) = ...