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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7
votes
2answers
210 views

Find $\lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n dx$

Find: $$\lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n dx$$ This is one of the problems i have to solve so that i could join college. I tried using integration by parts, i tried using ...
0
votes
0answers
9 views

Divergence Theorem: Conditions for the boundary integration to vanish?

Consider the Divergence Theorem for example in two dimensions, in the upper right quadrant of Euclidean space: $$\int_0^\infty dx \int_0^\infty dy ~\vec\nabla\cdot\vec F=\oint_C ds~\vec n\cdot\vec F$$...
1
vote
2answers
46 views

Differentiation under the integral sign in $R^3$

I'm trying to take derivative from an integral. I know about the Reynolds transport theorem, but I do not know how to obtain the unit normal and the velocity. I'm going to take the derivate from the ...
1
vote
1answer
80 views

Certain type of integrals $ \int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}}, $

I would like to do the following integral $$ \int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}}, $$ where the $i\epsilon$ has been added to avoid some possible divergencies. ...
0
votes
1answer
25 views

PDE with a condition

Considering the heat equation, $$\frac{du}{dt}=\frac{d^2u}{dx^2}$$ if $$u(x,t)=t^{\alpha}\phi(\xi)$$ with $$\xi=x/\sqrt{t} \enspace then \enspace \phi \enspace satisfies \enspace \alpha\phi-(1/2)\xi\...
2
votes
0answers
66 views

Calculating the Integral of a non conservative vector field

I have no clue how to do part C because a) is non conservative What I got for b) $f(x,y)=\dfrac{x^3}{3}+2yx+\dfrac{y^3}{3}+K$ (I don't know the symbol for the thing so I used f(x,y) instead. How do I ...
1
vote
1answer
393 views

Using Stokes' Theorem to evaluate an integral around a triangular path

Problem: Use Stokes’ theorem to evaluate the integral $I = \int\limits_C \textbf{F} \centerdot \textbf{ds}$ when $\textbf{F}$ is the vector field $\textbf{F} = 3zx\textbf{i} + 3xy\textbf{j} + yz\...
3
votes
1answer
68 views

Another way to evaluate $\int\frac{\cos5x+\cos4x}{1-2\cos3x}{dx}$?

What I've done is this:$$\int\dfrac{\cos5x+\cos4x}{1-2\cos3x}{dx}$$ $$\int \dfrac{\sin 3x}{\sin 3x}\left[\dfrac{\cos5x+\cos4x}{1-2\cos3x}\right]{dx}$$ $$\dfrac {1}{2}\int\dfrac{\sin 8x -\sin 2x +\sin ...
12
votes
1answer
660 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
1
vote
1answer
31 views

Integral of a line with random gradient

Consider a random line $Y = Mx$ where $M$ is a standard normal variable $M \sim \mathcal{N}(0,1)$. The line is integrated between 0 and 1: $$I = \int_{0}^{1} Y dx = \int_{0}^{1} Mx dx$$ What is the ...
30
votes
1answer
569 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}xdx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
0
votes
1answer
16 views

double integration changing variables and finding boundaries

This may be quite simple, but I am a little rusty and have had quite a few attempts at this and would appreciate some help. The form of the integrand does not really matter, I am trying to change the ...
0
votes
0answers
40 views

Integral of following fractional part

What is the integral of \int {nx} where {.} is a fractional part and $n\in R$ the integration range is from $-4$ to $4$.I want to use it in one of problems of the ellipse. Thanks . I haven't shown any ...
1
vote
2answers
42 views

$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
2
votes
2answers
126 views

Good books on integrals [duplicate]

I'm a math student at the sixth semester and I've had my courses in calculus and complex analysis. I'm able to solve integrals with the usual techniques. However, whenever I am confronted with a ...
1
vote
2answers
61 views

Understanding how to calculate surface area of parametrized surfaces

I am trying to follow a derivation for surface area of a parameterized surface and my book does not explain the reasoning behind different steps. I understand the derivation for surface area for a ...
4
votes
0answers
110 views

Integral of combination of power, exponential, and confluent hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(gx-b)^{2}}\,e^{-\beta_{1}x}\, {_{1}}F_{1}(-\alpha_{1};-\alpha_{3};\beta_{3} x) \ \mathrm{d}x \end{...
1
vote
0answers
23 views

Integrated Brownian Bridge is a Gaussian Process

Let $W(t),t \in [0,1]$ be a (Standard) Wiener Process. The Brownian Bridge $B(t), t \in [0,1]$ can be constructed via $B(t):=W(t) - t \cdot W(1)$ and is a Gaussian process with zero mean and ...
1
vote
0answers
14 views

Does make sense define a gauge for the integral $\int_2^x\frac{\sum_{n\leq t}\Lambda(n)}{t}dt$, where $\Lambda(n)$ is the von Mangoldt funtion?

I try encourage to me to study and understand the definition of gauge integral. See for example this reference Schechter, The Gauge integral where is explained the definition with an example. It is ...
2
votes
2answers
77 views

Evaluate $\int \frac{x}{x^4+5}dx$

$$\int \frac{x}{x^4+5}dx$$ $u=x^2$ $du=2xdx\Rightarrow \frac{du}{2}=xdx$ $$\int \frac{x}{x^4+5}=\frac{1}{2}\int \frac{du}{u^2+5}$$ I want to get to the expression in the form of $\frac{da}{a^2+1}$...
5
votes
3answers
94 views

Evaluate $\int \frac{\sin^4 x}{\sin^4 x +\cos^4 x}{dx}$

$$\int \frac{\sin^4 x}{\sin^4 x +\cos^4 x}{dx}$$ $$\sin^2 x =\frac{1}{2}{(1- \cos2x)}$$ $$\cos^2 x =\frac{1}{2}{(1+\cos2x)}$$ $$\int \frac{(1- \cos2x)^2}{2.(1+\cos^2 2x)}{dx}$$ $$\frac{1}{2} \int \...
1
vote
2answers
93 views

Evaluate $\int\sin^{7}x\cos^4{x}\,dx$

$$\int \sin^{7}x\cos^4{x}\,dx$$ \begin{align*} \int \sin^{7}x\cos^4{x}\,dx&= \int(\sin^{2}x)^3 \cos^4{x}\sin x \,dx\\ &=\int(1-\cos^{2}x)^{3}\cos^4{x}\sin x\,dx,\quad u=\cos x, du=-\sin x\,...
0
votes
1answer
66 views

Continuity of a function on a square

Fix some $\ell\in\mathbb{R}^+$, and say I have a function $f:[0,\ell]\times[0,\ell]\to\mathbb{R}$ with the following properties: $f(s,t)$ is continuous everywhere except when $s=t$, where it is ...
5
votes
1answer
30 views

Limit of measures, two questions on limits of integrals

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Call the limit $\mu(A)$. I ...
5
votes
4answers
216 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $

Let $\gamma$ be the Euler-Mascheroni constant. I'm trying to prove that $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $$ I tried introducing a parameter to the ...
0
votes
2answers
32 views

Changing Integral Bounds

I'm studying for Exam P and I was wondering what the need to change the lower limit of this integral was. Substituting $u=1+x$, $du=dx$, $$ \int_0^\infty\frac{3x}{(1+x)^4}\;dx = \int_1^\...
-1
votes
2answers
129 views

how is it that $\int_0^x 2\pi r\ dr$ is equal to the area of a circle [closed]

I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen ...
1
vote
2answers
71 views

Dominance between two functions

Let two functions $f(z)$ and $g(z)$ with $z\in[0,c]$ with $c$ a constant such that $c<b$. I'd like to check whether $f(z)-g(z)>0$. I've tried to set $f(z)$ to its minimal value and $g(z)$ to its ...
1
vote
0answers
17 views

Centroid of circle intersection

A region $R$ is bounded by the circle radius $4$ centred at $(5,0)$, the circle radius $2$ centred at $(0,0)$ and the x-axis, as shown. What is the centroid of this region? Finding the top ...
1
vote
1answer
45 views

Integral of product of a sigmoid function and beta distribution over (0, 1)

I am trying to solve the following integral, $$\int_0^1 \frac{1}{1+e^{-x}}\cdot Beta(x|\alpha, \beta)dx$$ where $Beta(x|\alpha, \beta)=\frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}$ is a ...
4
votes
2answers
75 views

Indefinite integral of a rational function with linear denominator: $ \int \frac{ x^7}{(x+1)}{dx} $

$$ \int \frac{ x^7}{(x+1)}{dx} $$ $$ \int \frac{ \left(x^7 + x^6 - x^6 - x^5 + x^5 + x^4 -x^4 - x^3 + x^3 + x^2 - x^2 -x^1 + x^1 +1 -1\right ) }{\left(x+1\right)}{dx}$$ $$ \int { \left(x^6 - x^5 + x^...
0
votes
4answers
39 views

fundamental calculus theorem application

I'm not sure about the use of the Theorem. I have: $$f(x)=\int_0^{x^2}(t-1)g(t)dt$$ I need the derivative of $f$. I know i have to apply the chain rules, but i'm not sure about the results. My result ...
0
votes
2answers
43 views

I need help understanding The Integral Test for series

For the following Series I have to show that the series qualifies for The Integral Test, then use it to determine if the series converges or diverges. here's my work where I apply the Integral Test, ...
0
votes
1answer
33 views

Show that $T(x+1)=xT(x)$ if $T(x)=\int _0 ^\infty e^{-t}t^{x-1} dt$

I need to calculate the following: the integral is from 0 to infinity I am given that $ T(x) = \int \limits _0 ^\infty e^{-t}t^{x-1} dt$. I need to show that $ T(x+1)=xT(x) $. So my logic was to ...
1
vote
2answers
80 views

What is $\int_{-1}^{1} x^2 d(\ln x)$

How to deal with the following integral: $$\int_{-1}^{1} x^2 d(\ln x)$$ It immediately struck me that why not put $d(\ln x) = \dfrac{dx}{x}$ and simply integrate, so we get answer as zero. Other way ...
6
votes
1answer
53 views

Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$

There are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, ...
0
votes
0answers
17 views

Hypocycloid with an outer ellipse

I have tried to change the traditional hypocycloid a bit. What I've basically done is that a circle now rolls inside an ellipse. I am trying to find the equation for the same. I am mostly done, ...
0
votes
0answers
19 views

When are these integrals equal to each other?

Suppose I have the integral: $$ F = \int^1_0 \, \int_{x \in C} L[u + t(v -u)] (v - u) \, \mathrm{d} x \, \mathrm{d} t $$ where $L$ is an operator of type $(\mathrm{R} \rightarrow \mathrm{R}) \...
4
votes
0answers
178 views

Is this function increasing?

I'm stuck in showing that the following function is increasing over the domain $\left[0,x_0\right]$: $$\Pi\left(z\right) = \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}}\left(2y-b\left(x\right)-x\...
1
vote
1answer
62 views

Help on line integral $\int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$

I need help on the following line integral: $$\int\limits_\gamma f dz = \int\limits_\gamma \frac{1}{g} dz = \int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$$ For a fix $r \in ...
0
votes
1answer
58 views

Integrate function by partial derivative

I'm searching a $\phi(x,t)$ solution of a pde cauchy system, with $x\in[-1,1],t\in[0,T]$ I am able to know: a) $\phi(x,0)=-cos\left(\pi\left(x-0.85\right)\right)$ b) $\phi_x(x,t)$, $\forall t,x$ (...
1
vote
0answers
28 views

Can I apply integration by parts to the integral $\int_{-\infty}^{\infty}\left[u'(x)|_{x=a_0}\right](x-a_0)v(x)dx$

Suppose, I have an integration $I=\int_{-\infty}^{\infty}u(x)v(x)dx$, where $u:X \to Y$ and $v: X\to Y'$ are $n^{th}$ order differentiable functions of $x$. Expanding $u$ around an arbitrary point $...
2
votes
1answer
87 views

integrate $\int e^{\sqrt{x}}dx$

$$\int e^{\sqrt{x}}dx$$ $t=\sqrt{x}$ $dt=\dfrac{dx}{2\sqrt{x}}\Rightarrow dx=2\sqrt{x}\,dt\Rightarrow dx=2t\,dt$ $$\int e^{\sqrt{x}} \, dx=2\int e^t t\,dt$$ $u=t\Rightarrow u'=1$ $v'=e^t\...
4
votes
0answers
36 views

Integration of Hilbert space valued mappings.

TL;DR: Is there a version of the Bochner integral which allows for the integration of isometric embeddings $\phi:X\to H$ from a metric space to a Hilbert space, satisfying $\int_X \|\phi\| d\mu < \...
-1
votes
1answer
48 views

Method of integration [duplicate]

We have to find the integration of the following function I tried but got stuck can anybody help me how to proceed . Is there anyother method to solve this
0
votes
1answer
39 views

Integrals in Reverse

I'm asked what solid this integral represents(the integral is used to obtain the volume of a solid, we are given this). I see that since we have the $ 2\pi$, this is probably a volume obtained by ...
-4
votes
0answers
44 views

Indefinite Integral of $\frac{1}{1+x^8}$ [closed]

How to find the indefinite Integral of $\dfrac{1}{1+x^8}$ by hand? Thanks for help. Please show the steps and the method used.
-4
votes
0answers
14 views

Integration problems The curve in the picture shown has equation y = bx(x − 2)… [closed]

I really need urgent help for this and I would like to thank people for your help in advance. Thank you. The problem set is here. Just click it! Integration
2
votes
2answers
39 views

Fundamental theorem of calculus statement

Let f be an integrable real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by F(x)=$\int _{a}^{x}\!f(t)\,dt$ Doesn't this make F(a) = $0$ ?
41
votes
6answers
3k views

How can this function have two different antiderivatives?

I'm currently operating with the following integral: $$\int\frac{u'(t)}{(1-u(t))^2} dt$$ But I notice that $$\frac{d}{dt} \frac{u(t)}{1-u(t)} = \frac{u'(t)}{(1-u(t))^2}$$ and $$\frac{d}{dt} \frac{...