Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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4
votes
5answers
98 views

integrate $\int \frac{dx}{x\sqrt{1-x}}$

$$\int \frac{dx}{x\sqrt{1-x}}$$ $$\int \frac{dx}{x\sqrt{1-x}}$$ $u=1-x$ $du=-dx$ $$-\int \frac{du}{(1-u)\sqrt{u}}$$ $a(1-u)+b\sqrt{u}=1\Rightarrow a-au+b\sqrt{u}=1$ $a=1\Rightarrow ...
1
vote
2answers
64 views

Integrating $\int^1_0 \dfrac{x^2e^{\arctan x}}{\sqrt{x^2+1}}$

This is a very hard integral that I am trying to solve. I’ve tried many substitutions, integration by parts, but I cannot evaluate this. Are there any other approaches I can take to solve this ...
7
votes
1answer
70 views

Closed form for Euler sum with $H_{2n}$?.

I ran across this Euler sum while trying to evaluate an integral. I mentioned it in another thread, but though perhaps asking about it separate may be a good idea. Is there a closed form for this ...
0
votes
3answers
23 views

How to account for solids of revolution around vertical lines to the right of the x axis?

I'm trying to find the volume of a solid created by rotating the region enclosed between $x=y^2$ and $x=1$ around the line $x=8$. Noting that the intersections of the functions occur at $(0,0)$ and ...
1
vote
0answers
12 views

reduction formula for $\int \tan^n (2x)dx$

Establish a reduction formula for $$\int \tan^n (2x)dx$$ My attempt, Let $I_{n}=\int \tan^n (2x)dx$ $=\int \tan^2 (2x) \tan^{n-2} (2x)dx$ $=\int (\sec^2 (2x)-1)\tan^{n-2}(2x)dx$ $=\int ...
0
votes
1answer
16k views

Finding the median value on a probability density function

Quick question here that I cannot find in my textbook or online. I have a probability density function as follows: $\begin{cases} 0.04x & 0 \le x < 5 \\ 0.4 - 0.04x & 5 \le x < 10 \\ ...
2
votes
3answers
48 views

Calculate $\int_0^1 \ \int_0^1 \ x \sin \lvert x^2-y^2 \lvert \; dx \; dy$

$$\int_0^1 \ \int_0^1 \ x \ \sin \lvert x^2-y^2 \lvert dx \ dy $$ $$\int_0^1 \frac{1}{2} \Big[ \sin \lvert x^2-y^2 \lvert \Big]_0^1 \ dy= \int_0^1 \frac{1}{2} \Big( \sin \lvert 1-y^2 \lvert - ...
1
vote
0answers
11 views

Prove that the condition $x(\tau)>\xi$ of a divergent integral implies that $x(t)>\xi$

Let $ E, J \subset \mathbb R$ be open intervals and let functions $h:J \to \mathbb R$ and $g: E \to \mathbb R$ be continuous. let $\xi \in E$ and assume that $g(\xi)=0$. Define $f:J \times E \to ...
3
votes
0answers
45 views

How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...
2
votes
2answers
46 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
4
votes
3answers
126 views

How do I show that if $f$ is bounded and integrable on $\mathbb{R}$, then $g(t) = \int_t^{t+1} f(x) dx$ is continuous?

Usually functions of the form of $g(t)$ tell me I should use the fundamental theorem of calculus, but I don't think that applies here because I'm not given that $f$ is continuous. I know from the ...
0
votes
6answers
62 views

Simple integration just learning this application

I want to integrate a function as $f(x)=\sin^{-1}x$. What should be the proper method of doing it?
0
votes
1answer
32 views

Finding Value of C to Maximize Area

f(x)=$xe^{-\sqrt x}$ Find the value of c, such that the area bounded between the graph, the x-axis, x=c, and x=c+1 is maximized. Find the maximum area. I don't know where to start with this one. I ...
2
votes
0answers
26 views

Imaginary number and absolute value integral - Fourier transform

I came across this integral problem: $$\hat f(\xi)=\int_{-\infty}^{+\infty} e^{-|x|+xi\xi}dx$$ Now I know how to integrate simple absolute value functions like: $\int_{-2}^{4}|x-2| dx$, we just ...
0
votes
1answer
22 views

Trouble getting between steps when solving integral

I've having a lot of trouble trying to figure out how they're getting from the step in blue to the one in red. Can some one please explain that?
1
vote
1answer
27 views

Density of $L^\infty(\Omega)h$ in $L^p(\Omega)$ where $h \in L^p(\Omega)$

Let $(\Omega,\mu)$ be a finite measure space. Suppose $1\leq p <\infty$. Let $h$ be an element of $L^p(\Omega)$ with $h >0$ a.e.. How show that the subspace $L^\infty(\Omega)h=\{ f h\ :\ f\in ...
0
votes
3answers
24 views

How to find differentiation and integration of curves in general?

Graph of function $f(x)$ How do I go about finding integration and differentiation of curves like these which yield other curves?
2
votes
1answer
169 views

Another way of doing integration

What's your option for calculating this integral? No full solution is necessary, it's optional as usual. Calculate $$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) ...
0
votes
1answer
40 views

Indefinite trignometric integral

I tried $u$-substitution and $uv$-substitution, can't seem to figure this out... any help would be appreciated! Question: $$\int\frac{x}{\cos(x)}\,dx$$ Thanks!!!
0
votes
1answer
15 views

Vector integral in $N$ dimensions.

In $N$ dimensions I want to do an integral of the flux through an $N-1$ dimensional surface. The usual vector calculus integration theorems help by allowing integration around the perimeter of the ...
-3
votes
1answer
72 views

How to solve this definite integration [on hold]

$$ I = \int\limits_0^\pi \frac{d\theta}{\left[(\alpha - \beta \cos \theta)^2 + c \right]^2 + d^2} $$ Source. I'm looking for a numerical method/scheme which can be used to solve the following ...
0
votes
0answers
71 views
-1
votes
0answers
37 views

Why can triple integrals be used to represent the volume under a surface by using $f(x,y,z) = 1$ in the integral?

I read that triple integrals can be used to represent the volume under the surface providing that we set the integrand to $1$, that is, $$ \iiint f(x,y,z)\,dx\,dy\,dz $$ where $f(x,y,z) = 1$. Why ...
2
votes
1answer
33 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
2
votes
1answer
19 views

Volume of region in the first octant bounded by coordinate planes and a parabolic cylinder?

Find the volume of the solid region in the first octant bounded by the coordinate planes, the plane $y + z = 2$ and the parabolic cylinder $x = 4 - y^2$. I have a final answer, I would just like to ...
0
votes
0answers
16 views

Estimate integration

Suppose $n<<k$ and $n<i<k$. How can I estimate this integral: $\int_{1}^{k-i} x^{-0.4}*(x+i-1)^{-0.4}dx$. I would like to get the result in the form of O(f(k,i)). Since the integrand is ...
1
vote
2answers
44 views

Finding the volume bounded by a cylinder and a plane

I have been given the following equations: $$x^2 + z^2 = 9$$ $$ x = 0 $$ $$ y = 0 $$ $$ z = 0 $$ $$ x + 2y = 2 $$ and have been asked to find the volume of the bounded region. I understand the ...
1
vote
2answers
78 views

Solving $\int_0^{\pi/2} \sin^2\theta \sqrt{1-k^2\sin^2 \theta}d\theta $

I have the following integration to solve. $$f(k) = \int_0^{\pi/2} \sin^2\theta \sqrt{1-k^2\sin^2 \theta}d\theta,\quad0<k<1$$ assuming $\sin\theta = t$ which results $d\theta = ...
3
votes
1answer
44 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
1
vote
2answers
66 views

Integrate $\int \frac{x^7}{(1-x^4)^2}dx$

$$\int \frac{x^7}{(1-x^4)^2}dx$$ I have tried to simplify the expression, to use U substation, any idea where to start from?
-6
votes
2answers
201 views

How does the continuity of a composite function of floor change when it is integrated, but with floor treated like a constant?

I'm doing some pretty strange integrals (floor functions ones) and I think I should probably start asking some more complex questions regarding it. Since I now know how to integrate them, I have to ...
1
vote
0answers
29 views

Is there a close form expression for the integral $ \int_a^b |x-c|^n e^{-x^2/2} $

Is there a close form expression for the integral \begin{align} \int_a^b |x-c|^n e^{-x^2/2} dx \end{align} by close form I mean it can be in terms of well know functions such as $Q$-function, ...
1
vote
2answers
46 views

Arc length of Archimedes Spiral $ r = \theta $ from $ 0 \le \theta \le 2\pi$

The equation of the Archimedes spiral is given by $$r = \theta$$ The formula for calculating the Arc Length is given by $$L = \int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$ The ...
0
votes
2answers
59 views

How do I evaluate a series? [on hold]

In this specific example, I don't understand the steps of evaluating this series: \begin{align} &\frac{12}{n}\left(\left[\sum_{i=1}^n-7\right]+\sum_{i=1}^n\left[\frac{-12}{n}\cdot ...
16
votes
8answers
700 views

A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$

How to prove the following $$\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx=\frac{\pi^2}{2}$$ I thought of separating the two integrals and use the beta or hypergeometric ...
5
votes
4answers
3k views

Computing $ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $

I would like to compute: $$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $$ We have: $$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx=2\int_{0}^{\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx$$ ...
2
votes
0answers
35 views

Asymptotic behaviour of an integral depending on a parameter

I am trying to compute the asymptotics on $t$ of the following integral: \begin{equation} I(t)=\int_{\mathbb{R}^{n}}e^{-|\lambda|^{2}/2t}\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} ...
1
vote
7answers
95 views

Integrate $\int \frac{x\cos x}{\sin^2x}dx$

$$\int \frac{x\cos x}{\sin^2x}dx$$ $$\int \frac{x\cos x}{\sin^2x}dx=\int \frac{x\cos x}{1-\cos^2x}dx=\int \frac{x\cos x}{(1-\cos x)(1+\cos x)}dx$$ How can I find the two fractions? if there are ...
-1
votes
1answer
17 views

how to prove if $f$ is integrable and $f'$ too then the limit of $f$ is zero when $x$ go to infinity? [on hold]

If $f$ is a real function on $\mathbb R$ and we have $\int_1^\infty |f(x)|dx < \infty$ and $\int_1^\infty |f'(x)|dx < \infty$ then $\lim_{x\to\infty}f(x)=0$ ?
1
vote
0answers
40 views

Compute $\frac{d}{dt}\int_0^t e^{x(s)}ds$, where $x$ is a standard Brownian motion.

How to compute the following differentiation? Is there a general rule that can be applied? $$\frac{d}{dt}\int_0^t e^{x(s)}ds$$ in the case of $x=W$ where $W$ is a standard brownian motion, is there ...
-2
votes
2answers
77 views

Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$ [on hold]

As a consequence of this Q, I need some help evaluating the following integral: $$\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$$ Integration by parts wouldn't simplify things and I guess that ...
1
vote
2answers
31 views

Finding distance of $h(t)=t$ from a closed subspace $Y$ of $\pi$-periodic functions in $L^2(-\pi,\pi)$

Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$ Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$ where $h(t)=t$. Compute ...
0
votes
1answer
35 views

finding area using iterated integral

I am trying to find the area enclosed between $f(x)=\sin x$ and $g(x)=\cos x$ between $x= \pi/4$ to $x = 5 \pi/4$. I got $\int_{\pi/4}^{5\pi/5}\int_{\cos x}^{ \sin x} dydx$. But I am not getting the ...
0
votes
1answer
53 views

Regarding Apostol's theory of integration

I have some questions regarding the theory of integration as discussed in Tom Apostol's Calculus. Integration is defined using step functions. My question is, is this definition he presents equivalent ...
1
vote
3answers
24 views

Proof of integral involving the inverse hyperbolic secant and cosent

We know that $$ \int \frac{dx}{x \sqrt{a^2 \pm x^2} } = -\frac{1}{a} \ln \frac{a+ \sqrt{a^2 \pm x^2}}{\lvert x\rvert }+C$$ I tried proving this integral setting $x = a \ \mathrm{csch} \ u $ and using ...
2
votes
2answers
36 views

Advanced calculus, Riemann integral.

If $f$ is (Riemann) integrable on $[a,b]$ and if $\int_{a}^{b} fh=0$ for all continuous function $h$, then $f(x)=0$ for all points of continuity of $f$. I know if we have $f$ being continuous on ...
4
votes
1answer
39 views

How would you integrate this?

If we had the following integral: $$\int_{a}^{b} {\big(1+x^2 \big)^s} \space dx$$ Where $s$ is not given. Is there any general formula for this integration that works for all $s\in \mathbb{R}$?
3
votes
1answer
30 views

Trigonometric integrals and limits

Show $$\lim_{N\to\infty}g_N(\theta_N)=2\int^\pi_0\frac{\sin x}{x}dx-\pi,$$ where $$g_N(\theta_N)=\int_0^{\theta_N}\frac{\sin[(N+1/2)x]}{\sin(x/2)}dx-\pi,$$ $$\theta_N=\frac{\pi}{N+1/2},$$ and ...
3
votes
2answers
83 views

How to solve integrals where you can't factor a polynomial?

Hi there guys I don't know if the title of the question should be the one for this but the thing is that I'm trying to solve this integral $\int \frac {\frac 12-u^2}{2u^4-2u^2+1}$$du$ and I have this ...
5
votes
1answer
91 views

Improper integral: $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $.

mathematica is reporting that the improper integral $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $ coverges to $2\cos(1)$. However, when I try to confirm this by actually integrating it using ...