Questions about the evaluation of specific definite integrals.

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Having Troubles With This Integration Problem

The question I'm having troubles with is as follows: Evaluate $\int_{-r}^r\sqrt{r^2-t^2}\,dt$ (Hint: substitute $t=rsinx$) So, immediately I did $dt=rcosx\,dx$ and substitute it all in... $\int ...
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Leibniz integral rule definition

https://en.wikipedia.org/wiki/Leibniz_integral_rule If we have an integral $$\int_{y_0}^{y_1} f(x, y) \,\mathrm{d}y$$ then for $x$ in $(x_0, x_1)$ the derivative of this integral is thus ...
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Relationship between Lippmann-Schwinger integrals of different dimensions

Define $G_n (\mathbf{x},\mathbf{x}')$ as $$ G_n (\mathbf{x},\mathbf{x}') = \lim_{\epsilon \to 0^{+}} \left[\dfrac{1}{(2 \pi \hbar)^{n}} \int_{\mathbb{R}^{n}} \mathrm{d}^{n}\mathbf{p} \dfrac{e^{i ...
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I am a Math Hobbyist. I have made some simple discoveries in Math. How do I share it with the Math community out there? [on hold]

I am a Computer Engineering graduate and have taken many courses in Math of course. While I was in the University, I got myself lost in the world of mathematics and I discovered stuff that I felt ...
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1answer
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Finding volume of a solid of revolution

I need to find the volume of the solid that is formed when the (x>0, y< -1) region of the curve y= -1/x is rotated about the y-axis. If I'm correct, this volume can be calculated by: Evaluating ...
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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 ...
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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 ...
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1answer
43 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!!
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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 ...
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437 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
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1answer
65 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 ...
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5answers
102 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
102 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? ...
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2answers
64 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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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 ...
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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 ...
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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 ...
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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?
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1answer
73 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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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 ...
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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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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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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
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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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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
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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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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
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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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Difference between definity integral with constants and with variables

What is general difference between: $$\int_a^b f(x) \;\mathrm{dx}$$ $$\int_a^{x^2} f(x) \;\mathrm{dx}$$
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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.
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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 ...
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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 ...
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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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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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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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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?
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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 $$\frac{x^2e^x}{x^4} - \frac{2xe^x}{x^4}$$ $$x^{-2}e^x - 2x^{-3}e^x$$ $$ ...
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2answers
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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 ...
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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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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
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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 - ...
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3answers
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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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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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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 ...
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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 ...
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2answers
77 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 ...
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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 ...