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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0
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1answer
14 views

$\int_C (\alpha x, -\alpha y) . dr = 0$ where C is the unit circle

Circulation is given by $$\int_C u . dr$$ I want to show that the circulation around the unit circle is $0$ for $u = (\alpha x, \alpha y)$. Ie. $$\int_C (\alpha x, -\alpha y) . dr = 0$$ How would ...
0
votes
2answers
90 views

Evaluating $\int \frac {\ln(3x+7)}{x^2}\mathrm dx$

How to evaluate the following integral? $$\int \frac {\ln(3x+7)}{x^2}\,\mathrm dx$$ I tried substituting both $u = 3x+7$ and $u = \ln(3x+7)$, but the resulting integral seems to be much more ...
1
vote
1answer
57 views

Convergence of series by using counting measure

Problem; Let $\{a_n\}$ and $\{r_n\}$ be two sequences of real numbers such that $\displaystyle\sum_{n\geq 1} |a_n|<\infty$. Prove that $$\sum_{n\geq 1} \frac{a_n}{\sqrt{|x-r_n|}}$$ converges ...
0
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2answers
60 views

Evaluate $\int_3^9 \frac{1}{x \log x} \,\mathrm dx$

$$\int_3^9 \frac{1}{x \log x} \,\mathrm dx$$ I tried : $$u=\log x \implies x \ln 10 du = dx$$ $$\ln 10 \int_{\log 3}^{2\log 3} \frac{1}{u} du = \ln 10 \left[\ln u \right]_{\log 3}^{2\log 3}=\ln 10 ...
8
votes
3answers
237 views

Evaluate $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$

Evaluate $$\displaystyle\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$$ How do I evaluate this integral? I know that the result is $0$, but I don't know how to obtain this. Wolfram|Alpha ...
1
vote
1answer
46 views

Region integral of exponential function

I am trying to evaluate the following integral $$\int_R \exp\left({\frac{x^2}{16}+\frac{y^2}{9}}\right)\,\mathrm dA$$ with the boundary conditions: $$\frac{x^2}{16}+\frac{y^2}{9}=2$$ ...
0
votes
4answers
82 views

How to integrate $\int_0^x \frac{1+\epsilon X}{1-X}~dX$?

I'm having a hard time trying to figure out the steps to get to the final answer shown below. $$ \int_0^x\frac{1+\epsilon X}{1-X}dX = (1+\epsilon)\ln\frac{1}{1-X}-\epsilon X $$ Any help would be ...
3
votes
2answers
105 views

Evaluate $\int_0^{\pi}\frac{x\sin x}{1+\cos^2x}\,\mathrm dx$

I can't seem to get any kind of result which I find useful when I use the fact that $$\int_0^af(x)\,\mathrm dx=\int_0^af(a-x)\,\mathrm dx$$ After using trigonometry, I end up getting: ...
1
vote
3answers
84 views

Integrate $\int_0^\pi e^{-ik\cos\theta}\sin^2{\theta} \, \mathrm{d}\theta$

Does anyone have some good method to integrate $$\int_0^\pi e^{-ik\cos\theta}\sin^2{\theta} \, \mathrm{d}\theta$$
1
vote
1answer
113 views

Fatou: Reverse?

Attention The usual problems are about absolute convergence: $$\int|g_n|\mathrm{d}\mu\quad(g_n=f_n,f-f_n,s_m-s_n,\ldots)$$ (There Fatou may help out!) But as proceeding with Fatou one encounters ...
2
votes
1answer
58 views

Vector field with gradient and integral over curve

The problem is: Consider the vector field: $$\textbf{F}= 4x^3y^3 \,\textbf{i} + (1+3x^4y^2) \,\textbf{j}$$ a) Find a potential function $ϕ(x,y)$, i.e. a function $ϕ(x,y)$ such that $\nabla ϕ= ...
0
votes
1answer
199 views

Work required to pump water out of tank in the shape of a paraboloid of revolution

This is the problem I have been assigned: A water tank has the shape of a paraboloid of revolution: its shape is obtained by rotating the parabola $y=x^2/4$, for $0\le x\le 4$, around the ...
2
votes
1answer
81 views

Question about length of curve?

The question: Find length of curve defined by $\displaystyle y=2\ln\left[\left(\frac{x}{2}\right)^2-1\right] $ from $x=4$ to $x=6$ Here is the work I have done, but I seem to keep getting it ...
0
votes
1answer
57 views

Find $\int_2^{2.2}f(x)\,\mathrm dx$ given $f(x)=x^4-3x^3+9x^2+22x+6$.

$f(x)=x^4-3x^3+9x^2+22x+6$. Find $\int_2^{2.2}f(x)dx$ by finding $f(x-2)$ This is in a non-calculator paper which is why $f(x-2)$ is meant to be obtained (it's supposed to made the maths possible to ...
0
votes
1answer
44 views

Evaluating $\int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm dx$ [duplicate]

$$\int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm dx$$ I am having trouble parsing the square of sine in the numerator. Could someone provide some hint? Thanks.
5
votes
2answers
83 views

Evaluate $\int_0^1 \frac1 {x^2+2x+3}\,\mathrm dx$

I first completed the square: $$\int_0^1 \frac1 {2+(x+1)^2}\,\mathrm dx$$ Made the substitution $x+1=\sqrt2 \tan u$. Thus $dx=\sqrt2\sec^2udu$ substituting this in and changing the limits (please ...
1
vote
0answers
40 views

Stuck on an integral of form $\int\exp(-\frac{\alpha}{m^2} - \beta m)\frac{dm}{m}$. Any ideas?

My statistical model involves the multiplication of a scalar random variable $X|X \geq 0 \sim 2\mathcal{N}(x;0,\sigma^2)\ \mathbb{I} \ [x \in \mathcal{R}_+]$, or a gaussian variable that must be ...
0
votes
0answers
34 views

Using Frobenius Theorem to solve a fairly general system of 1st order PDEs

It has been proved that if $\mathbb{X}$ and $\mathbb{Y}$ be the vector fields on $\mathbb{R}^3$ given by $\mathbb{X}(x,y,z)=(1,0,p(x,y)r(z))$ $\mathbb{Y}(x,y,z)=(0,1,q(x,y)r(z))$ ...
2
votes
1answer
104 views

Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= \left(\frac{\partial}{\partial t}\hat{\Phi}_t \right) \hat{\Phi}_t^{-1} \\ &= \left(\frac{\partial}{\partial ...
4
votes
1answer
298 views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
1
vote
1answer
68 views

Differential Equation help: $\frac{dy}{dx}=\frac{y-3}{x^2 +y^2}$

The question is: solve for $y,$ $\dfrac{dy}{dx}=\dfrac{y-3}{x^2 +y^2}$ given it passes through $(0,1)$. I am struggling to find a way to separate the variables. Also as a side question, if you have ...
4
votes
1answer
47 views

When is an oscillating integral small?

I hope, the title is not too confusing. My question is the following: We all know the Riemann-Lebesgue-Lemma stating that for $f\in L^1(\mathbb R)$, one has $$ \lim_{k\to\infty} \int ...
1
vote
0answers
26 views

Modifying python code to get imaginary solutions of numerical integration method

I stumbled upon this code to integrate functions using the numerical method Gauss Kronrod which I have been recommended for the functions I am integrating. So far in all the trials I have used I have ...
0
votes
1answer
70 views

Marginal of Dirichlet distribution is Beta (integral)

Just for the sake of simplicity, take $K=3$ then a random vector $(X_1,X_2,X_3)$ has a Dirichlet distribution, i.e. $(X_1,X_2,X_3)\sim Dirichlet(\alpha_1,\alpha_2,\alpha_3)$ if the density takes the ...
1
vote
1answer
58 views

On consequences of $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$

If $f : [0,1] \to \mathbb R$ is a continuous function and $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$ then is it true that i) $\int_{0}^1(f(x))^2dx=0$ ? ii) ...
0
votes
1answer
76 views

Find the mass of a sphere with density given by $\rho(r,\theta,\phi)$

The density is given by $$\rho=\rho_0e^{-r/R}(1-\cos\theta)$$ Where $R$ is the radius of the sphere. I integrate as follows: So, first integration is $$\int_0^R \rho_0 e^{-\frac{r}{R}} (1-\cos ...
0
votes
4answers
54 views

Show that $\int_0^a f(x)dx=\int_0^a f(a-x)dx$ [duplicate]

I don't really know where to start with this one. Can you just ignore the $f(..)$ and deal exclusively with what's inside the brackets?
3
votes
1answer
61 views

I don't understand a proof about volume and surface of revolution

So I am investigating volume and surface of revolution, in particular the shape called Gabriel's Horn, which is $$\int_1^\infty \frac{1}{x}dx.$$ The interesting property about this shape is that it ...
0
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1answer
37 views

Finding volume by integration

Well, the question says "The area bounded by hyperbola $xy=4$ and line $x+y=5$ is revolved about $x$-axis. Find the volume of solid thus formed." Having known that this site doesn't solve your ...
2
votes
1answer
35 views

Estimation of a sum independent of $n$

Suppose $f$ is differentiable on $[0,1]$, $f(0)=f(1)$, $\int_0^1 f(x)dx=0$, $f'(x)\neq 1$. Furthermore, let $g(x)=f(x)-x$, $n\geq 2$ is an integer. Show that ...
4
votes
2answers
78 views

Evaluate $\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$

$$\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$$ Please guide me by which term it should be substituted to get the result of this integration. I have tried it by using $\cos x =t$, but it went so long ...
0
votes
0answers
23 views

How to integrate concavity as a function of itself

For an equation such as the following $\dfrac{d^2z}{dt^2} = kz$ How would I find z in terms of t? I've tried multiplying both sides by $\dfrac{dz}{dt}$, but I'm not too sure how that really helps. ...
3
votes
1answer
233 views

Evaluating $\int \arccos\bigl(\frac{\cos(x)}{r}\bigr) \, \mathrm{d}x$

The title says it all, really - I am looking for $$\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$$ where $0<r<1$ and $x$ is in a domain where the integrand is real. It came up ...
0
votes
1answer
39 views

Area of a region bounded by $y=\sqrt{|x|}$ and $5y = x+6$

Find the area of the region bounded by $y=\sqrt{|x|}$ and $5y = x+6$ by looking at where the curves intersected on a graph I got $$\int_{-1}^4\Bigg[\frac{x+6}{5} - \sqrt{|x|}\Bigg]\,\, dx + \int_4^9 ...
1
vote
0answers
55 views

How to prove $\oint \frac{\pi\csc(πz)}{(2z+1) \cosh\left(\frac{\pi}{2} (2z+1)\right) } \,\mathrm dz ≈ 0$

In proving the $\displaystyle\frac{200}{\pi} \sum_{k=0}^\infty\frac{(-1)^k}{(2k+1) \cosh\left((2k+1) \frac\pi2\right) }= 25$ I use the below integral, because in calculation the integral by the ...
2
votes
0answers
111 views

How to show that a piecewise constant function is integrable, using the upper and lower sums?

Let $f(x) = \begin{cases} 1 &\mbox{if } 0\leq x<1 \\ 3 &\mbox{if } 1\leq x<2 \\ 2 &\mbox{if } 2\leq x\leq 3. \end{cases}$ Show that $f(x)$ is integrable by $(a)$ ...
1
vote
2answers
100 views

Evaluating Integral involving exp

I am stuck at the following integral :- $$ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-x^2-\frac{1}{x^2}\right)\,dx$$ Can anybody give me some hint. and also for this function $$ \int_{- ...
0
votes
2answers
50 views

Is it true that the probability $\mathbb{P}(S > t)$ is equal to $\int_0^t \mathbb{P}(S > x) \, dx$?

I have a question about the following equality $$\int_0^t \mathbb{P}(S > x) \, dx = \mathbb{P}(S > t),$$ where $S$ is a positive random variable. I can't see why this equality should hold. It ...
0
votes
1answer
28 views

1st order differential equation with x(t) and y(t) in one equation

$y$ and $x$ are all in term of $t$ but after I have found the integrating factor and multiply is to the both side, then RHS will become $xe^{Rt/L}$ and don't know how to continue integrating the ...
0
votes
1answer
35 views

An integral yielding dirac delta (edited)

Is the equation below true for $ x>0$ ? $$\lim_{x\rightarrow0}\left(\int_0^\infty \exp(-Rx)\cos(R(y-t)) \, dR\right)=\pi \delta(t-y)$$ Actually i don't understand why the multiplier $\pi$ exists ...
1
vote
0answers
26 views

Can an integral in the exponent of an exponetial function be written as a product?

I am asked to simplify/calculate the following integral: $$\frac{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] \right)|u(q)|^2}{\int d^3q \exp \left( -\beta C \int d^3q [u(q) u(-q)|q|^2] ...
3
votes
1answer
120 views

Describing non-vanishing $1$-forms on two dimensional manifolds.

Let $h_1 \mathrm{d}x_1 + h_2 \mathrm{d}x_2$ be a non-vanishing $1$-form on a $2$-dimensional manifold. Why do locally exist smooth functions $f,g$ with $f\mathrm{d}g= h_1 \mathrm{d}x_1 + h_2 ...
1
vote
3answers
66 views

Evaluating $\int{\frac{du}{3e^{u}+1}}$

why is $$\int{\frac{du}{3e^{u}+1}}=\ln\frac{e^u}{3e^u+1}+c$$ ? I think some substitution should help solving this integral, but everything I tried did not work.
0
votes
1answer
43 views

Integrate $\int_{p_1}^{p_2}(\frac {\partial G}{\partial p})_T$ with respect to p given that G is a state function

The equations discussed below apply to thermodynamics but the question is mathematical: $$dG=(\frac {\partial G}{\partial p})_T+(\frac {\partial G}{\partial T})_p$$ The above is an exact differential. ...
2
votes
2answers
63 views

boundedness of integral of a bounded function

Let $a(t)$ be a bounded function. Is $\int_0^xa(t)dt$ also bounded for all $x\in\mathbb{R^+}?$
4
votes
1answer
61 views

How can I find the Cauchy Principal Value of this integral using complex analysis?

I'm supposed to solve the real integral using a contour integral (The Cauchy Principal Value). Can someone give me a hand? I cannot seem to be able to do it... This is what I've tried so far: I ...
0
votes
1answer
117 views

integration, laurent series, residue therorem

Evaluate the integral $\int_\gamma f(z)dz,$ where $\gamma(t)=e^{it}$, and $0\leqslant t\leqslant2\pi$. For $f(z)$ equal to: $$\dfrac{e^z}{z^3},\quad\dfrac1{z^2\sin z},\quad\tanh ...
0
votes
1answer
37 views

Stokes Theorem, Evaluating the Integral with z<1 over a cylinder

First I find the intersection that is $z=1$ and parametrize it: $r(t)=1cost(i)+1sint(j)+1(k)$ $r'(t)=-sint(i)+cost(j)$ I then substitute this into $\int_C \! F(r(t)\cdot r'(t) \, \mathrm{d}t.$ to ...
3
votes
0answers
44 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
2
votes
3answers
92 views

Evaluating $\int_{0}^{1} \left ( \ln \frac{1}{x} \right)^ndx$.

I have found this result on the Internet $$\int_{0}^{1} \left ( \ln \frac{1}{x} \right)^ndx = \Gamma(n+1).$$ I know that if $n \in \mathbb{N}$, the proof is not complicated. However, if $n \in ...