All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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Banach Spaces: Improper Riemann Integral

Disclaimer This thread is related to: Stone's Theorem Definition Given a measure space $\Omega$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Denote the measurable subsets of finite ...
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3answers
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Example of Riemann integrable $f: [0,1] \to \mathbb R $ whose set of discontinuity points is an uncountable and dense set in $[0,1]$

Give example of a function $f: [0,1] \to \mathbb R $ which is integrable ( Lebesgue or Riemann , if possible , both) but whose set of discontinuity points is an uncountable set and dense in $[0,1]$ ...
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Solving second order ordinary differential equations

I would like you to ask you to solve the following two ordinary differential equations. It is highly appreciated your kind consideration in advance. 1.Solve y"(x)=f(x) 0<=x<=1, where f is ...
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prove integration formula relating to derivatives

Could any one help me solve this problem ? it is from Apostol's calculus volume 1
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Hard sum with harmonics numbers

Prove or disprove $S=\sum_{n=1}^{\infty}\frac{{H_n^{2}}{H_n^{(2)}}+3{H_n^{(4)}}}{n2^n}=\frac{25}{16}\zeta(5)+\frac{7}{8}\zeta(2)\zeta(3)$
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$ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $?

I have the following implication, and I need to determine whether it's true: $ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $ I tried solving ...
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$F(x) = \int_0^x f(t)~dt \implies F(1)=f(0)+\int_0^1(1-t)f'(t)~dt$?

f is differentiable and has a continuous derviative, and $F(x) = \int_0^x f(t)~dt$. Based on this assumption, I have the following statement which I need to determine whether it's true or false: ...
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f is even or odd, prove that f^2 is even

I need to verify whether a statement is correct or false. The statement is as following: If the function f is either odd or even, then the function f^2 is even. To my understanding, the statement is ...
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19 views

big $\mathcal O$ for number of prime in an interval?

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function $$\pi(x)=Li(x)+\mathcal O(\sqrt x \log x)$$ I am trying to understand how to deal with the ...
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Integrals: Average(f)*Average(g)=Average(f*g) [on hold]

So I've got everything but question #3 here. I understand that it isn't simply (1/4)(1/4)=16. And also not (1/4)(1/4)(1/4)=1/64. But I can't think of what else it might be. It isn't discussed in the ...
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Evaluate $\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$ using residue method [on hold]

This is a real integral but I want to evaluate it using residue integration method $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$$
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Integration by Substitution, can't solve (Working Added )

My Working: $$\displaystyle dx = du/2x$$ sub dx and U into equation $$ x^2 \int x(U)^{3/2} du/2x$$ Eliminate x $$ x^2/2 \int (U)^{3/2} du$$ $$ x^2/2. [2(U)^{5/2}/5]$$ then $$ ...
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4answers
63 views

Evaluating $\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$ using trigonometric substitution [on hold]

Using Substitution Integral Method, compute $$\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$$ (let $x=2\cos^2\theta+5\sin^2\theta$)
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Recursive formula for Laguerre guassian integral?

The integral of interest is: $ I_{l, m} = \int_{u0}^{u1} u^{(l+1)/2} e^{-u/2} L_m^l(u) du $ where $L_m^l$ is the laguerre polynomial. What I'm interested in is getting some relation to lower order ...
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3answers
225 views

Indefinite integral of a simple function

$\int 2(1 + \tan^2 x)$ My work : $2(1 + \tan^2(x) = 2 + 2\tan^2x$ $2x + \frac{2}{3}$ $\tan^3(x) \cdot \ln|sec(x)| + C$ The answer says no, after multiple tries :(
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1answer
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Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
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4answers
57 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
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1answer
13 views

Singular chain complex for integration - pinching on boundary

Singular chain complex, as far as topology are concerned, is just continuous map from standard simplex, and the choice of using simplex over other shape is immaterial. But for integration on manifold, ...
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1answer
65 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
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2answers
75 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
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4answers
49 views

Integration by parts of $\cos(x)e^{-x}dx$

I do the integral but I end up getting the original $\cos(x)e^{-x}dx$ on both sides and canceling them out resulting in no solution. Can I get a step by step break down of how to solve?
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How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
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1answer
30 views

Problem with this question on solid of revolution

Calculate the volume of a revolution solid obtained by rotation around the x-axis, the region bounded by the graph of $y=e^x$, $-1\le x \le1$ and the x-axis. Thanks in advance, and sorry about my ...
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1answer
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Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition, a nonexample and a comparison see: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample ...
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Simple indefinite integral of a vector function

I am having trouble with this simple integration. I am not sure of the process or steps to follow to solve this type of problem: If $\mathbf{V}(t)$ is a vector function of $t$, find the indefinite ...
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When substituting in integration, do you have to change the limits of integration so long as you keep it consistent?

I have this integral: In order to solve for it, I have to substitute: t=tan(theta) dt=(sec(theta))^2 d(theta) When substituting that, I know I have to change the limits of integration within ...
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42 views

Piecewise vs Continuous Integration

I have the following data: Daily spend on marketing Daily gain of fans because of that spend on marketing ('billed' fans) The 'organic' daily number of fans for the same period above (ie free ...
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45 views

Integration of a polynomial

I am facing a problem in finding the integral $$\int\frac{r^2}{-C r^3 + r^2 -2 M r +Q^2}\,dr$$ Here M, Q, and C are parameteres (to be fixed later). Could anybody Please help me in finding it? I ...
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2answers
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Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
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1answer
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Uniform convergence and integrability

If $(f_n)_{n \in \Bbb N}$ converges to $f$ uniformly and each $f_n$ integrable would it imply $f$ is integrable and $$\lim_{n \to \infty}\int f_n = \int f$$ In case each $f_n$ is nonnegative ...
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3answers
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Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
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1answer
27 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
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3answers
58 views

Problems with this integral $ \int \sqrt{1 + {1 \over t^2} + {2 \over t}} dt$

$$ \int \sqrt{1 + {1 \over t^2} + {2 \over t}}\,\mathrm dt$$ I tried making substitution, using $ u=1 + \dfrac{1}{ t^2} + \dfrac{2 }{ t} $, then , $dt=\dfrac{du}{-2\left({1 \over t^3 }+ {1 \over ...
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Evaluating $\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} dy$

I was trying to find arc-length of $x = \sqrt{y}-7y$ So basically right now I am stuck with this $$\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} \,\mathrm dy$$ $$\int^{4}_{1} ...
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2answers
27 views

Arc length of a curve which already has an integral sign

This one here was tricky because the function already has an integral sign. My guess is that I need to evaluate the indegral where $x=4$ so that i get $y=f(t)$ and after that apply the Arc Length ...
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2answers
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Integration $\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R$.

Give a compact form for the solution of integral: $$\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R,k\in\mathbb N$$ any suggestions please?
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1answer
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Partial Derivative of a nonexistant variable?

I am wondering how I would find the partial derivative of $z = g(r, \theta) = \theta$ with respect to both $r$ and $\theta$. I realize that if you take the partial in respect to $\theta$, it is 1. I'm ...
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0answers
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Antiderivative of $|x − 2| + |x − 3|$ [on hold]

Find the most general antiderivatives of the following function. $$|x − 2| + |x − 3|$$ I started with showing that the antiderivative for $|u|$ is $\dfrac{u|u|}2$. How to proceed then?
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The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
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1answer
51 views

Taking the Derivative of $F(x)=\int_0^x f(t)\,dt$ [on hold]

Let $F(x)=\int_0^x f(t)\,dt$ What is the derivative of $F(x)$? I desperately need guidance!
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1answer
34 views

Lebesgue integration in one variable

I have studying the conditions for the existence of the Lebesgue integral. Generally, to show that existence of the integral of a function on an unbounded interval, one can integrate and take ...
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1answer
30 views

Verify Green's Theorem for region bounded by the lines $x=2$, $y=0$, $y=2x$

Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, $y=2x$ and the functions $f(x,y)=(2x^2)y$, $g(x,y)=2x^3$. I have been trying this question for far too long and I can't ...
2
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1answer
26 views

Inequality involving Holders Inequalities

Suppose $f\in L^p(\mathbb{R})\cap L^\infty(\mathbb{R})$ for some $p>2$, show that $||f||_{p}\leq ||f||_2^{2/p}||f||_{\infty}^{1-2/p}$ I tried to write $|f|^p=|f|^{\frac{p}{2}}|f|^{\frac{p}{2}}$ ...
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1answer
25 views

Prove that the antiderivative of an integrable function is both bounded and integrable

Let $f: [a,b] \to \mathbb{R}$ be a bounded function which is also integrable. Define $F: [a,b] \to \mathbb{R}$ by $$F(x)=\int_{a}^xf(t)\ dt$$ To prove that $F(x)$ is also bounded and integrable I ...
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128 views

Show that $\displaystyle\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$

$$\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$$ My Intuition telling me there might be an $\arctan$ coming up, but I don't know how to do this ...
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A cylinder with base radius of 9 𝑐𝑚 and height 16 𝑐𝑚 is cut by the plane [on hold]

A cylinder with base radius of 9 𝑐𝑚 and height 16 𝑐𝑚 is cut by the plane 𝑥 + 𝑦 + 𝑧 = 9 and 𝑧 = 0. Using triple integral, find the volume of the cylindrical section bounded by the planes.
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Question on area under and between curves and volume of a solid by revolution

I have recently begun learning about finding the area under the curve by definite integrals. But I am still a little unsure of the concepts. When you integrate for a certain range of the graph , does ...
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1answer
22 views

Second mean value theorem proof

I am asked to prove the second mean value theorem: Let $f$ and $g$ be defined on $[a,b]$ with $g$ continuous, $f\ge 0$, and $f$ integrable. Then there is a point $x_0 \in (a,b)$ such that $$ ...
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1answer
15 views

ML-inequality for real integrals

For a homework assignment from my analysis class, I was asked to show the following: Let $f\colon [a,b] \to \mathbb{R} $ be Riemann integrable and $|f(x)| \le M$. Define $F(x) = \int_{a}^{x}f(t)dt$. ...
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2answers
32 views

When do you drop the absolute value from ln|x| + C when integrating $\frac{1}{u}du$

Given: p(t) represents the number of cats, when t>=0. Given: p(t) is increasing at a rate directly proportional to $800-p(t)$ So, I represent this as: $\frac{dp}{dt}= k(800-P)$ I want p(t), so I ...