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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2
votes
1answer
40 views

log-sobolev inequalities for infinite measures.

I was wondering if we could have log-sobolev inequalities for infinite measures, most notably Lebesgue measure. I presume this is false, but I haven't been able to construct one. I tried playing ...
3
votes
3answers
92 views

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ we have $$ \int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} \,dt = O(z^{-1} )? $$ According to Serge Lang, the integral on the left is the error term for ...
0
votes
0answers
48 views

Theoretical justification for separable differential equations using substitution.

Usually when I have solved separable differential equations, I have just followed a recipie and never understood why I am allowed to do what I do. I see that the heart in solving these equations is ...
1
vote
0answers
57 views

Shell Method About Y-Axis

In my calculus course, we just covered the Shell Method and its uses. I have been doing the homework for a few hours and I am absolutely stumped by a question. The question states: Find the ...
2
votes
0answers
91 views

Integral $\int^{1}_{-1} \frac{ln(ax^2+2bx+a)}{x^2+1}dx$ if $a>b>0$

I am trying to evaluate the following integral: $$\int^{1}_{-1} \frac{\ln(ax^2+2bx+a)}{x^2+1}dx,$$ where $a>b>0$. I can't really think of a way to find it so please give me a hint.
1
vote
1answer
36 views

Sum of continuous $L^{1}$ function over the integers.

Let $f$ be a continuous $L^1$ function defined on $\mathbb{R}$, such that $\hat{f}(k) = 0$ for $|k| > 1/2$, where $\hat{f}$ is the Fourier transform. Is it true that $\sum_{k = ...
0
votes
1answer
31 views

If $y=\int_0^x f(t)\sin[k(x-t)]dt$, then calculate $\frac{d^2y}{dx^2}+k^2y$

If $y=\displaystyle \int_0^x f(t)\sin[k(x-t)]dt$, then calculate $\dfrac{d^2y}{dx^2}+k^2y$ $y=\displaystyle \int_0^x f(t)\sin[k(x-t)]dt$ $\implies$ $\dfrac{dy}{dx}=\displaystyle \int_0^x ...
0
votes
3answers
212 views

Compute $\int\frac{1-x}{(x-2)(x+3)}$ and $\int\frac{cos(3x)}{sin(3x)}$

Compute $\int\frac{1-x}{(x-2)(x+3)}$ and $\int\frac{cos(3x)}{sin(3x)}$. I have no idea how to solve these 2 integrals, I've run out of ideas. The first one especially, I can't even start, not sure how ...
0
votes
2answers
47 views

Compute $\int_{1}^{e}\frac{1+\log x}{2x}dx$

I've been trying to solve this integral: $$\int_{1}^{e}\frac{1+\log x}{2x}dx$$ I used a new variable to solve this; $1+\log x = t$ therefore $dx = x dt$, then I inserted this into the original ...
0
votes
0answers
29 views

Periodic antiderivative

Let's consider an integral $\int{\ln(a \cdot \sin(x)+1) \cdot dx}=F(x)$. The value of $a$ is choisen so that $F(x)$ is a periodic function. Find $a$. Ideas: If $F(x)$ is periodic with a period $T$, ...
3
votes
0answers
49 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
0
votes
1answer
56 views

Computing the integral of $\int \frac{25x^2}{(x+3)(x-2)^2}\,dx$ [closed]

How would I find the indefinite integral of the expression $$ \int \frac{25x^2}{(x+3)(x-2)^2}\,dx $$ I have tried using impartial differentiation but was unsuccessful.
1
vote
0answers
40 views

Radial function and integral

Let $\Phi:(0,\infty)\rightarrow(0,\infty)$ be an increasing function and $\rho:(0,\infty)\rightarrow(0,\infty)$ is a function satisying the property $$ \frac{1}{C}\leq\frac{\rho(s)}{\rho(r)}\leq C ...
3
votes
4answers
89 views

Compute $\int_{-\pi}^\pi{x^2\sin\frac{x}{2}dx}$

I have to solve this integral: $$\int_{-\pi}^\pi{x^2\sin\frac{x}{2}\mathrm dx}$$ I'm thinking about integration by parts, but I'm not sure how to either derivate or integrate the $\sin\frac{x}{2}$ ...
6
votes
2answers
102 views

Asymptotic form of the integral $\int_{0}^{\infty} dx ~ \sqrt{x^2 + wx} ~ e^{-ixs}$ for $s \to \infty$

I would like to find an asymptotic form of the following integral when $s \to \infty$ ($s$ and $w$ are positive) \begin{equation} \int_{0}^{\infty} dx ~ \sqrt{x^2 + wx} ~ e^{-ixs} \end{equation} I ...
0
votes
2answers
72 views

How do I calculate $ \iiint_D|z|\,dx\,dy\,dz$ without using spherical coordinates?

I have the following integral: $$ \iiint_D|z|\,dx\,dy\,dz $$ which I need to integrate over the set: $$ D = \{x,y,z \in \mathbb{R}: x^2 + z^2 \leq y^2, y^2 \leq 4 \} $$ I have a problem ...
11
votes
3answers
174 views

How can I show that $ \int_0^\pi \frac{x\,dx}{1+\cos^2(x)} = \frac{\pi^2}{2\sqrt{2}} $

Show that $$ \int_0^\pi \frac{x\,dx}{1+\cos^2(x)} = \frac{\pi^2}{2\sqrt{2}} $$ I tried using change of variable $x = \pi-y$ and then ended up with integral $\int_0^\pi \frac{1}{1+\cos^2(y)}dy$ which ...
-2
votes
1answer
60 views

Does $\int_0^1 \sqrt{\frac{1+x}{\sin{x}}}$ converge?

Does $$\int_0^1 \sqrt{\frac{1+x}{\sin{x}}}dx$$ converge? I have tried to substitiute $x$ in nominator as $\tan{x}$ and simlify it using trigonometric formulas, but the integral was still too ...
0
votes
1answer
64 views

Calculating flux through a square

I did not quite understand the latest lecture I've been to and would like a thorough explanation if possible. A field vector is given by $F=(\cos(xyz), \sin(xyz), xyz)$. Calculate the flux through a ...
2
votes
2answers
43 views

show $\int f_kd\mu\leq C$ for $f_k\geq0$, $\int fd\mu\leq C$

Let $(\Omega, \mathcal A,\mu)$ be a measure space and $f_k\rightarrow f$ a.e., $f_k\geq0$ and $\int f d\mu\leq C$ for some $C>0$. How can you show $\int f_k d\mu\leq C$ ? My attempt: I thought ...
5
votes
0answers
82 views

$f$ is Riemann integrable iif the set of discontinuous points of $f$ has Lebesgue measure zero

This is a well known result in mathematics, but it's my first time attempting to prove it. I'm following the second book of Analysis from Folland. Below are the notations used and the theorem, from ...
6
votes
1answer
36 views

Integral with absolute value of the derivative

I'm trying to estimate this integral $\int_0^1 t |p'(t)|dt$ using this value $\int_0^1 |p(t)|dt$; here $p $ is a real polynomial. This means, I am looking for an $M>0$ such that $$\int_0^1 |t ...
5
votes
0answers
104 views

Prove that $\lim_{x\to\infty} f(x) = 0$

Let $f:\mathbb{R}\to\mathbb{R}$ which is continuously differentiable ($f\in C^1$). Lets assume $\int_0^\infty f < \infty$ and $f'(x)$ is bounded. Show that $\lim_{x\to\infty}f(x) = 0$. All I could ...
0
votes
4answers
142 views

Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? [closed]

Does $\int_0^\infty e^{-x}\sqrt{x}dx$ converge? Thanks in advance.
3
votes
0answers
151 views

Prove $\int_{0}^{1} |\frac {f^{''}(x)}{f(x)}| dx \geq 4$ [closed]

I find an interesting theorem,but have no idea to prove it. $f(x) \in C^2[0,1]$ and $f(0)=f(1)=0$ , $f(x) \not = 0 \ \ , x\in (0,1) $ Prove that if $\int_{0}^{1} \bigl|\frac ...
1
vote
6answers
118 views

Does $\int_0^\infty \sin(x^{2/3}) dx$ converges?

My Try: We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$ Hence, the integral after substitution is: $$ \frac{3}{2} \int_0^\infty ...
1
vote
2answers
539 views

Explain why graph of f lies below the $x$-axis in interval $[4\pi/9,5\pi/9]$

$f(x)=(x+1)\sin(3x)$ Explain why the graph of f lies below the $x$-axis for values of $x$ in the interval $[4\pi/9, 5\pi/9]$ From what i know/understand I'd have to look at the function in two ...
0
votes
1answer
386 views

Help me complete finding the Reduction formula of $J_n=\tan^{2n} x \sec^3 x dx$?

Please don't mark my question as a duplicate of Find the reduction formula for the following integral. This question, was asked by a user and I was trying to answer this, I couldn't complete my work ...
4
votes
0answers
42 views

Compute the complex integration [duplicate]

Let, $f(z)$ be an analytic function. Then the value of $$\int_{0}^{2\pi}f\bigl(e^{it}\bigr)\cos t dt= ?$$ (a) 0 (b) $2\pi f(0)$ (c) $2\pi f'(0)$ (d) $\pi f(0)$. $\mathcal{My}{Attemt}:$ ...
1
vote
1answer
148 views

Find the reduction formula for the following integral.

$$ \mathrm{(b)} \quad\quad J_n = \int \tan^{2n}(x) \sec^3(x) \;\mathrm{d}x $$ I have no idea how to start this question when it comes to the reduction formula. I know there are some cases for when ...
7
votes
2answers
210 views

$\int_0^\infty\int_0^\pi\frac{k^2(e^{-it\sqrt{k^2+m^2}}-e^{it\sqrt{k^2+m^2}})\sin(\theta)}{e^{-ikx\cos{\theta}}\sqrt{k^2+m^2}}d\theta dk$

$$\int_0^\infty\int_0^\pi\frac{k^2\left(e^{-it\sqrt{k^2+m^2}}-e^{it\sqrt{k^2+m^2}}\right)\sin(\theta)}{e^{-ikx\cos{\theta}}\sqrt{k^2+m^2}}d\theta dk$$ I saw this Integral at Quora, and I have not ...
0
votes
2answers
201 views

Let f be a convex differentiable function. Prove that if u is any continuous function, then … [closed]

Let $f$ be a convex differentiable function. Prove that if $u$ is any continuous function, then $$\frac1 a \int_0^a f(u(t))dt \geq f \bigg(\frac1a \int_0^a u(t) dt\bigg) $$ I need insight on this ...
0
votes
2answers
76 views

Is every compact set rectifiable? example

Is every compact set rectifiable? The set is rectifiable iff it is compact and the boundary is of measure $0$ (This is stated as a theorem). Can I infer from this that every compact set is ...
0
votes
2answers
35 views

Find the area between these two functions using integration

The functions are $$ f(x) = \ln (x) $$ $$ g(x) =(\ln(x))^2 $$ Is there a simple way of finding area other than using the long method of integration by parts
0
votes
3answers
76 views

Let $f$ be a continuous function on $[a,b]$ such that $\int_{a}^{b}f=0$ Prove that there is a number $z$ in $[a, b]$ such that $f(z)=0$.

Let $f$ be a continuous function on $[a,b]$ such that $\int_{a}^{b}f=0$ Prove that there is a number $z$ in $[a,b]$ such that $f(z)=0$. Show by an example that the continuity assumption is necessary. ...
0
votes
2answers
86 views

Simplify ratio of integrals $\frac{\int f(x-t) t e^{-t^2/2} dt}{\int f(x-t)e^{-t^2/2} dt}$

I am trying to simplify the following expression: \begin{align*} \frac{\int_{-\infty}^\infty f(x-t) t e^{-t^2/2} dt}{\int_{-\infty}^\infty f(x-t)e^{-t^2/2} dt} \end{align*} by getting it in terms of ...
0
votes
0answers
28 views

Trying to understand how the trapezoidal rule applies to a derivation of Stirling's Approximation

I am reading through the wikipedia article on how to derive the Stirling's Approximation. The article applies the Trapezoidal Rule to get the following: $$\begin{align} \ln (n!) - ...
1
vote
0answers
32 views

Techniques for computing (approximate or exact) partial sums for functions

Clearly there are several ways of computing the partial sum formulas of many summations, but is there a technique that can compute any partial sum. For example with $\sum_{x=0}^{n} \frac{1}{x}$, ...
0
votes
3answers
77 views

Evaluate $ \int_{a}^{b}(A - f(x))dx$ where $A = [1/(b-a)] \cdot \int_a^b f(x)\,dx$

My solution: Using the definition of the integral, rewrite $f(x)$ in the expression $A = [1/(b-a)] \cdot \int_a^b f(x) \, dx$ as: $$A = \frac1{b-a} \sum_{i = 0}^{n \to +\infty} f(x)\frac{b-a}n$$ ...
1
vote
1answer
56 views

Is the following logic of simplifying the complicated expression correct?

Assume I have three variables $s, \omega,\gamma$ and define a function $G(.)$. Next consider the following function \begin{align} \prod_{{i>1}}\left(\int_{-\infty}^{+\infty}\Bigg(\frac{1}{1+s ...
1
vote
3answers
87 views

Integral $\int_1^2 1/(x^2 \sqrt{x^2+1}) \, dx$ [closed]

What is $\int_1^2 1/(x^2 \sqrt{x^2+1}) \, dx$? Trigonometric substitution keeps getting too messy.
0
votes
1answer
30 views

Shifting Velocity and Position functions

I'm given a function $A(t)$ that defines the acceleration of an object w.r.t. time $t$ and am tasked with finding the position function and velocity function for that object. Finding the functions ...
2
votes
1answer
23 views

Sequence of continuous functions convergent to $0$ with the integral equal to $1$

I am looking for a sequence of continuous functions $\{f_m\}$ defined in $A\subset\mathbb{R}$ with $\lim\limits_{m\to\infty} f_m=0$ such that $\int_A f_m \;d\mu=1$. The problem I have is with the ...
0
votes
0answers
42 views

Integral Involving a Radical by $\arccos\left(f(x)\right)$: Any help?

I have $$ \int{x\sqrt{1-\left(\frac{x}{2}\right)^2}\arccos\left({\frac{a+x^2}{bx}}\right)}\,dx, $$ and I have tried using differentiation inside the integral to no avail. Mathematica quickly fails but ...
1
vote
4answers
174 views

Integral of $\cos^4(2t)\,dt$ with bounds from $0$ to $\pi$

$$\int_0^\pi\cos^4(2t)\,dt=?$$ I have attempted this problem two different ways and got two different answers that are nowhere near the correct answer. Could you please show me detailed steps on how ...
0
votes
1answer
50 views

I need help to evaluate this definite integral. [closed]

I need help to evaluate this definite integral: $$\int_{-\pi}^{\pi} \frac{\sin nx}{(1+2^x)\sin x}dx$$ ...
0
votes
1answer
17 views

how should a unbounded integrable function be like on a bounded set?

actually, my first question is could it be unbounded near boundary and we redefine the value on the boundary. For example, function $f=1/\sqrt{x}$ on $(0,1]$ and $f=0$ if $x=0$. is it integrable? Is ...
0
votes
3answers
45 views

Local Max of an Integral

I'm having trouble with the following problem. $f(x)=\int_0^x \frac{t^2-4}{1+cos^2(t)}dt$ At what value of $x$ does the local max of $f(x)$ occur? I've tried just taking the integral then ...
0
votes
0answers
18 views

Conditions for a simple integral.

Here is an ordinary differential equation: $dV(t)/dt=-r \times V(t)+g \times I(t)$ Under what condition, and how can this equation be integrated as: $V(t)=\int_0^te^{-r}gI(x)dx$ The potential ...
3
votes
2answers
109 views

How to show that $ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \ln\left(\tan(x)+\tan\left(\frac{\pi}{6}\right)\right)\tan(x)\space dx=\frac{\zeta(2)}{6} $

I was trying to prove the well known result: $$ \sum_{k=1}^\infty \frac{1}{\binom{2k}kk^2}=\frac{\zeta(2)}{3} $$ and it came down to prove the following equation: $$ ...