Questions about the evaluation of specific definite integrals.

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Conjecture $\int_0^{1}\frac{{(\rm arcsin})^2({x^2})}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}{\pi^3}…$

$$I=\int_0^{1}\frac{{(\rm arcsin})^2({x^2})}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}{\pi^3}-\frac{\pi}{2}ln^2{2}-2{\pi}\operatorname\chi_{2}(\frac{1}{\sqrt{2}})$$ This result seems to me digitally ...
1
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0answers
23 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
3
votes
1answer
39 views

Compute $\int_0^1 \frac{ 1}{1 + x^{1/2}}\,dx$. [on hold]

Basically, the question is $$\int_0^1 \frac{1}{1+x^{1/2}}\,dx.$$ I have no idea how to approach this and have spent hours to no avail. Any help would be gladly appreciated. Thanks!!
0
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0answers
18 views

Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
7
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5answers
430 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
2
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1answer
53 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
2
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5answers
101 views

Quick integral question

Sorry about the formatting, but how would I go about this question: $$\frac{d}{dx} \int_{\cos x}^1 \sqrt{(1 + e)^t} dt$$ What I've learned in class is that the derivative of an integral is just the ...
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2answers
94 views

Difficult Integral $\int_0^{1/\sqrt{2}}\frac{\arcsin({x^2})}{\sqrt{1+x^2}(1+2x^2)}dx=$

I have a difficult integral to compute.I know the result, but need to know the method of calculation. How prove this result? ...
2
votes
2answers
62 views

Prove that $\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$.

Prove that $$\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$$ for all $k\in\mathbb N$ and $h$ even integer. I have tried with Induction Principle (for $h$) but without success.
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4answers
55 views

Give that $f$ is a decreasing continuous function and that $f(x+y) = f(x) + f(y) -f(x)f(y)$ and $f'(0)=-1;$ Then find $\int_{0}^{1}f(x)dx$

Give that $f$ is a decreasing continuous function and that $$f(x+y) = f(x) + f(y) -f(x)f(y)$$ and $f'(0)=-1;$ Then it is to be found what is $\int_{0}^{1}f(x)dx$ I am at a loss on how to approach ...
1
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2answers
34 views

meaning of definite integral

So to my knowledge a definite integral's significance is how it shows the "intensity" or area under the curve for a function. However, I am confused then why the definite integral for x from 0 to 1 ...
0
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0answers
19 views

Riemann-Stieltjes Integral Substitution

I want to prove $\int^b_a\,f(g(x))\,dg(x) = \int^{g(b)}_{g(a)}\,f(x)\,dx$ for all f continuous. Firstly, $\int^b_a\,f(g(x))\,dg(x) = \int^b_a\,f(g(x))g'(x)\,dx$, since g is continuous and ...
3
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0answers
78 views

tough definite integral: $\int_0^\frac{\pi}{2}x\ln^2(\sin x)~dx$

Any ideas on $\int_0^\frac{\pi}{2}x\ln^2(\sin x)\ dx$ ? Best numerical approximation I can get is $0.2796245358$ Is there even a closed form solution?
0
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1answer
70 views

Closed form for this integral $I=\int_0^{1}\frac{{\arcsin}({x^2})}{\sqrt{1-x^2}}dx$

I’m trying to find a closed form for this integral.Any help is appreciated.Thanks $$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
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0answers
35 views

Meijer function as a product of two Meijer's functions

I want to evaluate an integral $I_1$ defined in $Eq.(1)$ as \begin{align} I_1=\int_{0}^{\infty}\frac{x\exp(-\beta x)K_1(\alpha x)}{1+x}dx\tag{1} \end{align} Where $K_1(.)$ is modified first order ...
0
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1answer
42 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \to ...
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4answers
63 views

Using trig identities to evaluate $\int_{0}^{\pi/2} \sqrt{1-\sin x} \, dx$

Use the identities $$\cos 2x=2\cos^2 x -1=1-2\sin^2 x$$ $$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$ to help evaluate $$\int_{0}^{\pi/2} \sqrt{1-\sin x} \; dx$$ I've already done ...
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0answers
17 views

Can an iterated integral over a box R ={(x,y,z)|x∈[0,a], y∈[0,b], z∈[0,c]} be expressed in eight different ways?

this is my first time on stack exchange so sorry if I am not following any guidelines. I received this exact question on a midterm and answered yes, it is possible, which was considered wrong on the ...
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0answers
27 views

Integral in vector form

I have two vectors $x_{1}$ and $x_{2}$ and some function $f(x, k)$. So for example function $f$ can be evaluated at some point say $f(x_{1}, 10)$. Then I have an integral written ...
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4answers
98 views

I want to solve $\int \frac{2}{x^2(x^2+1)^2}dx$

I want to solve this primitive $$I=\int \frac{2}{x^2(x^2+1)^2}dx.$$ I substitute $u=x^2$ then, $$I=\int \frac{2}{x^2(x^2+1)^2}dx=\int \frac{du}{u^{3/2}(u+1)^2}=\cdots$$ How do I use partial fraction ...
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1answer
50 views

How do I evaluate $\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\lambda}} dt$ ??

How do you evaluate: $$\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\alpha}} dt = ??$$ Many thanks.
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5answers
78 views

How can I solve $\int \frac{3x+2}{x^2+x+1}dx$

I want to compute this primitive $$I=\int \frac{3x+2}{x^2+x+1}dx.$$ I split this integral into two part: $$\int \frac{3x+2}{x^2+x+1}dx=\int \frac{2x+1}{x^2+x+1}dx+\int \frac{x+1}{x^2+x+1}dx,$$ For ...
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0answers
47 views

Help evaluating the integral of $\frac{1}{1+\sqrt{1-x^2}}$ over $[0,1]$ [on hold]

Can someone help me with this please: $$ \int_0^1 \frac{dx}{1+\sqrt{1-x^2}} $$ I substituted putting $u= \sqrt{1-x^2}$ and I got to this $$ \int_0^1 \frac{u\,du}{(u+1)\sqrt{1-u^2}} $$ I don't know ...
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1answer
42 views

How to calculate $\lim_{x \rightarrow 0} \frac{\int_0^{G(x)} \arctan(s+2s^2) ds}{x^2}$ based on the following assumption?

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. To start this question, we need to integrate $\arctan(s+2s^2)$ but how do you do that? Then, what do we do ...
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0answers
18 views

Difference between definity integral with constants and with variables

What is general difference between: $$\int_a^b f(x) \;\mathrm{dx}$$ $$\int_a^{x^2} f(x) \;\mathrm{dx}$$
2
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1answer
32 views

Solve the improper integral: $\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$

I'm completely stuck on this one. I only know that it converges thanks to Wolfram, but I don't know how to evaluate it. $$\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$$ Thank you for the help.
2
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3answers
122 views

Why doesn't $\int_{-1}^{1}\frac{dx}{x} = \ln|x|\biggr\rvert_{-1}^{1} = 0$?

$1/x$ is an odd function, so it makes sense to me intuitively that the area would be $0$, and similarly I would expect that $\int_{-1}^{2}\frac{dx}{x} = \ln(2)$. Proof Wiki seems to confirm my ...
1
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1answer
62 views

Compute $\int_0^{\infty} Q_1(y,b) \frac{y}{\sigma^2} \exp{(-y^2/(2\sigma^2))} \, dy$

We know that the first order Marcum Q-function can be represented as $$Q_1(y, b)=\int_{b}^{\infty} x \exp{(-(x^2+y^2)/2)} I_0(y x) \, dx ,$$ where $I_0(\cdot)$ is the modified Bessel function of the ...
2
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2answers
25 views

Use the Fourier transform to find value of definite integral from negative infinity to infinity

Find the value of $\int_{-\infty}^{\infty} f(x) dx$, where $f(x)=sin(x)/(x^3+x)$. How do I go about solving this? I have tried to expand the sine part into complex exponentials to try and resemble ...
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3answers
40 views

Contour Integration: non-convergent integral

The question is $$I=\int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} dx$$ My attempt: $$I=-\frac{1}{4}\int_{-\infty}^{\infty} \frac{e^{2ix}-2+e^{-2ix}}{x^2} dx$$ $$I=-\frac{1}{4} \Big[ ...
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0answers
17 views

proofing pyramid volume formula using integration.

I'm studying the proof of pyramid volume formula using integration. In the video minute 12:10 when the professor already got the $S$ variable which is the side of the pyramid base he continues to do ...
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0answers
51 views

How to find $\int_0^1 {1\over x}\,dx$? [on hold]

How do you calculate this integral: $$\int_0^1 {1\over x}\,dx$$ , using the integral definition?
1
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1answer
50 views

Integral of $x^{-2}e^x$

This is the original problem. $$\int_2^1 \frac{x^2e^x - 2xe^x}{x^4}$$ My attempt at breaking it down $$\frac{x^2e^x}{x^4} - \frac{2xe^x}{x^4}$$ $$x^{-2}e^x - 2x^{-3}e^x$$ $$ ...
1
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2answers
39 views

How to find the area for the curve $y=\sin^3(2x)\cos^3(2x)$?

I could calculate the integration of this by substituting $u=\sin(2x)$ and could find one of the limits of integration which was $0$. However, I couldn't find second limit. The mark scheme says the ...
0
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1answer
48 views

Find dy/dx of Integral [on hold]

really stuck on this problem, my textbook doesn't have ANYTHING like it. The only instruction is to find the dy/dx of the interval: $$y=x\int_2^{x^2}\sin(t^3)\,\mathrm dt $$ Thanks for any help!
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2answers
42 views

Integral of exponential rational function

I'm asked to find $$\int_0^{\ln 2}{e^{2x}\over{e^{4x}+3}} \text{ d}x$$ I can't for the life of me figure out how to integrate this.
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1answer
89 views

Proving $\int_0^{\pi } f(x) \, \mathrm{d}x = n\pi$

I've been asked to show $$ \displaystyle \int_{0}^{\pi} \dfrac{2(1+\cos x) - \cos((n-1)x) - \cos((n+1)x) - 2\cos nx}{1-\cos 2x} \ dx = n\pi $$ The integrand simplifies nicely to $$\frac{\cos nx - ...
1
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3answers
56 views

How to integrate $\int_0^{2\pi} \frac12 \sin(t) (1- \cos(t)) \sqrt{\frac12 - \frac12 \cos(t)}\,dt$

How to integrate $$\int_0^{2\pi} \frac12 \sin(t) (1- \cos(t)) \sqrt{\frac12 - \frac12 \cos(t)}\,dt$$ I know the solution is $0$, but I don't know how one gets this.
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1answer
33 views

Prove that, for each $n$, $\int^{1}_{0} f_n(x)dx=\frac{1}{2}$

Problem: Define $f: [0, 1] \to [0, 1]$ by $f(x)= 2x$ for $0 \leq x \leq \frac{1}{2}$ and $f(x) = -2x+2$ for $\frac{1}{2} \leq x \leq 1$ Let, $f_1(x)=f(x)$ and $f_{n}(x) = f(f_{n-1}(x))$ for all $n ...
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0answers
20 views

Alternative way to compute gamma integral of Chi-squared distribution

I am developing a software with Java, which is not comfortable at all wih integral computations. So I must compute the following integral (I need it to compute Chi-Squared CDF): $$ ...
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0answers
93 views

Solve $\int_0^1 \int_0^{2\pi}\frac{ax-x^2\sin(\theta)}{\sqrt{a^2-2ax\sin(\theta)+x^2}}d\theta dx$

Solve $$\int_0^1 \int_0^{2\pi}\frac{ax-x^2\sin(\theta)}{\sqrt{a^2-2ax\sin(\theta)+x^2}}d\theta dx$$ This integral is from the following paper : Frictional coupling between sliding and spinning motion ...
2
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3answers
60 views

How to evaluate these two integrals about hyperbolic functions?

While I was calculating the two integrals below \begin{align*} \mathcal{I}&=\int_{0}^{\infty }\frac{\cos x}{1+\cosh x-\sinh x}\mathrm{d}x\\ \mathcal{J}&=\int_{0}^{\infty }\frac{\sin x}{1+\cosh ...
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0answers
24 views

integration of product two lower incomplete gamma function and exponential [closed]

i need helping to find the value of this integration : $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$ where all parameters are positive. Can anyone help me how ...
2
votes
2answers
76 views

Integral of $x^2 e^{-x^2}$

Like the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has left ...
2
votes
1answer
38 views

Integral of a trig function divided by the square root of a polynomial: $\int_a^b\frac{\sin x}{\sqrt{(x-a)(b-x)}}dx$?

I was trying to help some physics students with an integral on their homework and they've presented me with something that has me stumped. The integral they are working on is: $$\int_a^b\frac{\sin ...
0
votes
1answer
68 views

Lower integral of the sum of two functions isn't equal to the lower integral of each summed separately?

I'm trying to figure out the problem above and I know I need to show what I have done so far, but I'm not even sure where to begin.
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2answers
74 views

Finding the value of this integral $ \int_{-\pi/4}^{\pi/4}{ (\cos{t} + \sqrt{1 + t^2} }\sin^3{t}\cos^3 {t})dt$? [duplicate]

I stumbled on this problem and I'm stuck. Any hints on how I could approach evaluating this particular integral? $$ \int_{-\pi/4}^{\pi/4}{ (\cos{t} + \sqrt{1 + t^2}\sin^3{t}\cos^3 {t})\,dt}$$ What ...
4
votes
2answers
125 views

Evaluating $\int^{\pi}_0\arctan\left(\frac{p\sin x}{1-p\cos x}\right)\sin(nx) dx$ by differentiation under integral?

I saw that $$ \int^{\pi}_{0}\arctan \left(\frac{p \sin x}{1-p \cos x}\right) \sin(nx) dx=\frac{\pi}{2n} p^n $$ for $$p^2 <1$$ I tried to prove using differentiation under integral but got ...
3
votes
1answer
64 views

Integrate $\int_{-\infty}^\infty e^{\frac{-x^2}{a^2}} dx$ [duplicate]

How to integrate $\int_{-\infty}^\infty e^{\large \frac{-x^2}{a^2}} dx$ using substitution? I know that $\int_0^\infty e^{-x^2}dx = \frac12\sqrt{\pi}$.
2
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1answer
70 views

Evaluating $\int_{0}^{\log\phi}\frac{u^2(e^{2u}+1)}{e^{2u}-1}du$, when $\phi$ is the golden ratio

When I do $$e^u=x+\sqrt{x^2+1}$$ in (17) page 9, see here, after some computations then I obtain $$\frac{\zeta(3)}{10}=\int_{0}^{\log(\phi)}\frac{u^2(e^{2u}+1)}{e^{2u}-1}du,$$ where ...