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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3
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0answers
43 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 ...
2
votes
2answers
36 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 ...
1
vote
2answers
24 views

Application of Integral: Work [on hold]

A 360-lb gorilla climbs a tree to a height of 20ft. Find the work done if the gorilla reaches that height in 10 seconds.
0
votes
0answers
22 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 ...
0
votes
2answers
47 views

Finding the volume of revolution using the method of shells

I'm trying to find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=6x+7$ about $x$-axis using the shell method. I applied the method and I got $15864/5$ ...
2
votes
1answer
71 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
0answers
29 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 ...
3
votes
2answers
52 views

Asymptotic form of an integral

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
3answers
44 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
1answer
27 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 ...
1
vote
1answer
39 views

Finding original function of $1/x$ by using a step function

Is it technically possible to show, that the area under $1/x$ in the interval $[1,a]$ equals $\log a$ without using any kind of differentiation, but only step functions. I tried this, but without ...
3
votes
0answers
65 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
3
votes
4answers
117 views

Computing $\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$

I want to compute $\int_0^\infty u^{-1}(1-e^{\frac{-u^2 t}{2}})\sin(u(|x|-r))\,du$ and so ,as shown below, I want to compute $$\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$$ Attempt We split ...
0
votes
0answers
22 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 $T$, ...
0
votes
1answer
56 views

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

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.
3
votes
4answers
69 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 $\sin\frac{x}{2}$ ...
0
votes
0answers
18 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 ...
0
votes
2answers
57 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 ...
5
votes
0answers
82 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 ...
-2
votes
1answer
52 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
68 views

Help for Integral and evaluating - Eikonal equation

Hy guys I'm reading a paper of "Finding Exact Solutions to the Two- Dimensional Eikonal Equation" - E.D. Moskalensky. link for the paper: ...
5
votes
3answers
373 views

Computing the indefinite integral $\int x^n \sin x\,dx$

$\newcommand{\term}[3]{ \sum_{k=0}^{\lfloor #1/2 \rfloor} (-1)^{#2} x^{#3} \frac{n!}{(#3)!} }$ I am trying to prove that for $n \in\mathbb N$, $$ \int x^n \sin x \, dx = \cos x \term{n}{k+1}{n-2k} + ...
0
votes
1answer
14 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 ...
1
vote
1answer
46 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 ...
9
votes
4answers
201 views

Ways to prove $ \int_0^1 \frac{\ln^2(1+x)}{x}dx = \frac{\zeta(3)}{4}$?

I am wondering if we can show in a simple way that $$ I=\int_0^1 \frac{\ln^2(1+x)}{x}dx = \int_1^2 \frac{\ln^2(t)}{t-1}dt = \frac{\zeta(3)}{4}. $$ Because the end result is very simple, I suspect ...
0
votes
2answers
32 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
4
votes
1answer
20 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 ...
4
votes
0answers
59 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 ...
3
votes
2answers
81 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: $$ ...
0
votes
4answers
85 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.
1
vote
6answers
112 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 ...
0
votes
2answers
67 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 ...
1
vote
1answer
35 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 ...
-3
votes
1answer
45 views

$\int^\infty_{-\infty}\int^\infty_{-\infty} e^{-(3x^2+2\sqrt 2xy+3y^2)} dx \,dy$

Evaluate $$\int^\infty_{-\infty}\int^\infty_{-\infty} e^{-(3x^2+2\sqrt 2xy+3y^2)} dx \, dy$$ Please give some hints how to proceed.
1
vote
3answers
73 views

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

What is $\int_1^2 1/(x^2 \sqrt{x^2+1}) \, dx$? Trigonometric substitution keeps getting too messy.
3
votes
4answers
520 views

Finding the definite integral of a function that contains an absolute value

The integral in question is this: $\int_{-2\pi}^{2\pi}xe^{-|x|}$ My attempt: Since there is a modulus, we split it up into cases. I'm not really sure which cases to split it into, do I just ...
7
votes
4answers
151 views

Prove $\int_{-\pi}^{\pi}\sin \sin x \,dx=0$ without using the fact that $\sin(x)$ is odd.

Prove $$\large\int_{-\pi}^{\pi}\sin (\sin x) \,dx =0$$ without using the fact that $\sin(x)$ is odd. Computing this in wolfram says that it is uncomputable, which leads me to believe that the only ...
5
votes
3answers
123 views

Evaluate the integral $\int_0^{\pi/2} \sin (2n x) \tan x\, dx$

Is there a elementary evaluation of the integral $ \int_0^{\pi/2} \sin (2n x) \tan x\, dx, $ where $n$ is the natural number? This number is related to the Fourier sine coefficient for $\tan (x/2)$.
20
votes
2answers
601 views

Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$

Show that $$\begin{aligned} \int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx &= \frac{7\pi^3}{108} \\ \int_0^{\pi/3}x\log^2 \left(2\sin\frac{x}{2} \right)dx &= ...
0
votes
1answer
35 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 ...
0
votes
2answers
47 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 ...
1
vote
1answer
38 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 ...
4
votes
0answers
32 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) $\pi f(0)$. $\mathcal{My}{Attemt}:$ ...
0
votes
2answers
81 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 ...
2
votes
0answers
35 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
1answer
13 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 ...
0
votes
3answers
51 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
0answers
18 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!) - ...
0
votes
3answers
64 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
0answers
17 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 $\sum_{x=0}^{n} \frac{1}{x}$, ...