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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-4
votes
1answer
72 views

How prove this integral with the limit [closed]

Find the limit $$\lim_{x\to 1^{-}}\left(2\int_{0}^{x}\dfrac{\ln{(1-t)}\ln^2{(1+t)}}{1-t}dt-2\ln{2}\ln{(1-x)} ...
1
vote
2answers
358 views

Generalization of Fatou's lemma for nonpositive but bounded measurable functions.

Let $(f_n)^{\infty}_{n=1}$ be a sequence of measurable (not-necessarily $\ge 0$). Let $g \gt 0$ be a measurable function with $\int g d\mu < \infty$ (integrable) such that $f_n\ge -g$ a.e. relative ...
3
votes
2answers
87 views

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

I'd like to evaluate the following definite integral: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
2
votes
3answers
70 views

Calculate $\int_{S^2}\frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}}dS$ where $a^2+b^2+c^2<1$.

Let $a^2+b^2+c^2 < 1$ and $S^2$ be unit sphere in $R^3$. Calculate $$\int_{S^2}\frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}}dS$$ Let $(x,y,z)=(\cos\theta \cos\phi,\cos\theta \sin\phi, \sin\theta)$. ...
2
votes
2answers
83 views

Primitive of $\frac{3x^4-1}{(x^4+x+1)^2}$

How to find primitive of: $$\frac{3x^4-1}{(x^4+x+1)^2}$$ I am having a faint idea of a type which may or maynot be in the primitve, i.e.: $$\frac{p(x)}{x^4+x+1}$$ The problem is I am not getting an ...
15
votes
3answers
2k views

Shortcut/trick for integrating a factored polynomial?

If I'm integrating a factored polynomial, say $$\int{x(x+1)(x-2)(x+3)dx},$$ does some shortcut exist that keeps me from having to expand the polynomial? Currently, I'd just do all the multiplication ...
-3
votes
0answers
24 views

surface integral [closed]

for qiestion 2 casing design here, im guessing this is one kind of surface integral question, and x 2 +y 2 +z 2 =5 2 , thus z=25−x 2 −y 2 − − − − − − − − − − − √ =25−r 2 − − − − − − − √ ...
9
votes
2answers
514 views

Why do we need the absolute value signs when integrating the square of a function?

Why do we need the absolute value signs in the definition of square-integrable function? As seen below: $$ \int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty $$
1
vote
1answer
33 views

Proving that $ \frac{1}{n}\int_{-\infty}^{\sqrt{n}w}k(v/\sqrt{n})\phi(v)dv$ is $O(n^{-1})$

Suppose that $h:\mathbb{R}\to\mathbb{R}$ is infinitely differentiable. Define \begin{equation} k(w)=\left\{ \begin{array}{ll} \frac{d}{dw}\left(\frac{h(w)-h(0)}{w}\right)&w\neq 0,\\ ...
17
votes
1answer
415 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
0
votes
2answers
41 views

Evaluating the integral to find the expected value of the exponential random variable

I want to find the expected value of the exponential random variable. I have $E(X)=\int_{0}^{\infty }xae^{-ax}dx$. I use Integration By Parts (IBP). Let $v=-e^{-ax}\Rightarrow dv=ae^{-ax}dx$, and ...
3
votes
3answers
73 views

Find the limit and derivative of integral function.

$\psi_m(x)$ is defined as $$\int_0^{\ln|x|}e^{mt}\sin(t)^m\mathop{dt}$$ with $m$ a natural number greater then zero. Now the question is, does $\lim\limits_{x\to 0}\psi_m(x)$ exist. I've tried using ...
2
votes
1answer
87 views

How to find $\int_0^1 \frac {\mathrm dx}{\left \lfloor{1-\log_2(1-x)}\right \rfloor}$

We want to evaluate; $$\int_0^1 \frac {\mathrm dx}{\left \lfloor{1-\log_2(1-x)}\right \rfloor}$$ The $\left \lfloor{x}\right \rfloor$ is the floor function. I have made no progress so far.
1
vote
1answer
32 views

Continuity of measure and integration

Suppose that f is a measurable function $(\Omega, \mathfrak{F}, \mu)$ such that $\int_{A}f \, d\mu \geq 0 \forall A \in \mathfrak{F}$. Prove that $f \geq 0 \ \mu$-almost surely. Hint: Let $A_n = ...
10
votes
2answers
299 views

A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$ I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help ...
9
votes
2answers
229 views

Evaluate $\int_{0}^{\large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\frac{\ln{(\sin x)}}{\cot x}+\frac{\ln {(\cos x)}}{\tan x}\right)dx$

How do I find the value of this integral? $$I=\int_{0}^{\Large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\dfrac{\ln{(\sin x)}}{\cot x}+\dfrac{\ln {(\cos x)}}{\tan x}\right)dx$$ I tried ...
1
vote
1answer
24 views

Calculating moment of inertia about the $z$-axis of solid with constant density

I have the following math problem: Find the moment of inertia about the $z$-axis of the solid in the first octant that is bounded by the coordinate planes and the graph of $x+y+z=1$ if the density ...
0
votes
4answers
78 views

General form for $2\int_{0}^{\infty} \frac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} \mathrm dt$

I encountered this integral in physics-- $$2\int_{0}^{\infty} \dfrac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} \mathrm dt$$ I know for certain that $a,b \in \mathbb{R^+}$, $a>b$. $a$ and $b$ are independent ...
0
votes
1answer
39 views

How to solve this: $F(x)=(x+y)e^{x-y}$, Find $\frac{\partial F}{\partial x}$ and $\frac{\partial F}{\partial y}$ [closed]

How Can I solve $F(x)=(x+y)e^{x-y}$, Find $\dfrac{\partial F}{\partial x}$ and $\dfrac{\partial F}{\partial y}$
3
votes
0answers
50 views

Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-\exp(-x^2/a^2)}\exp((x-u)^2/2s^2)$$ wrt $x$, where $a$ and $u$ are constants
2
votes
1answer
54 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
0
votes
2answers
51 views

If the product $fg$ is Riemann integrable then are $f$ and $g$ individually integrable?

If $fg$ is integrable, does this imply that $f$ and $g$ are both integrable too? I don't need a proof, if someone knows please just say (this will help me understand a thing about Taylor's theorem).
2
votes
2answers
83 views

Indefinite integral of $\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}$

How do I find $$\int\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}\mathrm dx$$ I used partial fractions by breaking up $x^2 + 2x - 2$ into $(x+1)^2 - 3$ and split it into $(a+b)(a-b)$ but as ...
2
votes
1answer
61 views

Contour integral in complex plane (tricky)

Let U be a simply connected domain with a simple closed boundary curve C oriented anticlockwise, and define for all w ∈ C \ C $$ g(w)=\oint_C \frac{e^zdz}{(z-w)^2}$$ Find a formula for g(w) which does ...
24
votes
8answers
4k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
1
vote
2answers
45 views

Reversing the chain rule

I'm pretty new to calculus, but is there a way to reverse the chain rule so I can take the antiderivative of 1/(x^3+1) without using partial fractions?
0
votes
0answers
17 views

How to compute using integration the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around a unit circle?

How to compute the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around the unit circle centered at the origin using the methods of the integral calculus?
1
vote
1answer
35 views

Mass centrum for $y=\sqrt{\ln (x+1)}$

The region below $$y = \sqrt{\ln (x+1)}$$ and above the $x$-axis, $ 0 \leq x \leq 1$, is rotated about the $x$-axis. So I want to find the mass centrum for this solid. We know that for the ...
1
vote
1answer
1k views

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. (weird equation)

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five ...
2
votes
1answer
93 views

Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $

If $a,b$ are fixed points of $f$, then $$ \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $$ In the words of 2014 MIT Integration Bee Champion (Carl Lian), the above property ...
-3
votes
0answers
42 views

Below this limit of integration should be how to solve? [closed]

$$\mathop {\lim }\limits_{x \to {1^{\rm{ - }}}} \left\{ {2\int_0^x {\frac{{\ln \left( {1 - t} \right){{\ln }^2}\left( {1 + t} \right)}}{{1 - t}}dt} - 2\ln 2\ln \left( {1 - x} \right)\int_0^x ...
5
votes
5answers
603 views

Finding $\int e^{2x} \sin{4x} \, dx$

Finding $$\int e^{2x} \sin 4x \, dx$$ I think I should be doing integration by parts... If I let $u=e^{2x} \Rightarrow du = 2e^{2x}$, $dv = \sin{4x} \Rightarrow v = -\frac{1}{4} \cos{4x}$ $\int{ ...
5
votes
2answers
121 views

How to choose the integration method for integrals involving powers and quotients of trigonometric functions?

I need help on these three integrals. Any hints on which method to use are greatly appreciated. $$1)\ \int \frac{1}{\cos^4 x}\tan^3 x\mathrm{d}x$$ $$2)\ \int \frac{1}{\sin 2x}(3\cos x + 7\sin ...
2
votes
3answers
104 views

What does one mean when $\int \frac{\sin x}x$ doesn't exist?

Well I say that by taylor's expansion: $$\int\frac{\sin x}x=\int\frac{x-x^3/6+x^5/120+...}x=x-x^3/18+x^5/480+...+\mathbb{C}$$ It's another thing that there doesn't exists a closed form for the ...
5
votes
1answer
74 views

How to evaluate $\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$

How to evaluate: $$\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$$ I have done a substantial work on it: Let $x^5z^3=x^5-1$. So $$x^5(z^3-1)=1\implies ...
3
votes
4answers
71 views

Shorter way to integrate $\int \frac{x^9}{(x^2+4)^6} \, \mathrm{d}x$

$$ I=\int \frac{x^9}{(x^2+4)^6}\mathrm{d}x $$ Yeah I know, I can substitute: $$t=x^2+4\text{ or }2\tan\theta$$ So that: $$I=\frac12\int\frac{(t-4)^4}{t^6}\mathrm{d}t\text{ or } ...
1
vote
1answer
36 views

Double Integral $\int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{1+y^3} \mathrm{d}y\;\mathrm{d}x$

I am having trouble computing the double integral: $$ \int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{1+y^3} \mathrm{d}y\,\mathrm{d}x $$ I computed the inner integral: $$ \left [ \frac{1}{3}\ln|y + 1| - ...
0
votes
1answer
101 views

How to integrate over curved extrusion paths?

Given a two-dimensional area $A\subset\mathbb R^2$ lying in the $xy$-plane, i.e. $A:\mathbb R^2\supset U\to A\subset \mathbb R^2, (u,v)\mapsto (x(u,v),y(u,v))$, a straight extrusion along the $z$-axis ...
4
votes
3answers
154 views

Geometric interpretation of an integral inequality

Let $f: [a, b] \to \mathbb [0, \infty)$ be an integrable function. By Cauchy-Schwartz: $$ \left(\int_a^b f(x) dx\right)^2 \leq (b-a) \int_a^b f(x)^2 dx$$ with equality iff $f$ is constant. If we ...
6
votes
1answer
131 views

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
0
votes
3answers
132 views

What are other unexpected results of integration?

I have integral of $\dfrac{1}{t^2 + 1}$ and integral of $\dfrac{t}{t^2 + 1}$ whose output is $\arctan(t)$ and $\dfrac12\ln(t^2 + 1)$ respectively. Are there any similar unexpected results when we ...
0
votes
0answers
35 views

Change of variables in 3 dimensions

I'm confused about a basic mathematical step. Say we have $$\omega=\frac{2\pi|\vec n|}{L}.$$ Then why is it that $$\int^\omega d^3\vec n= 4\pi L^3\int_0^\omega\frac{d\omega}{(2\pi)^3}\omega^2$$ ? ...
4
votes
4answers
318 views

Integral of exponential with $x(1-x)$ term

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\,-\sqrt{\,x(1-x)\,}\,\right)\,{\rm d}x $$ Or must I settle with an approximation? Edit: Actual form of integral ...
3
votes
2answers
214 views

How prove this integral $ \int\limits_0^1{\int\limits_0^1{\ln\Gamma\left({x+{y^3}}\right)}}dxdy =-\frac{7}{{16}}+\frac{1}{2}\ln 2\pi$

show that $$ I=\int\limits_0^1{\int\limits_0^1{\ln\Gamma\left({x+{y^3}}\right)}}dxdy =-\frac{7}{{16}}+\frac{1}{2}\ln 2\pi$$ where $$\Gamma{(a)}=\int_{0}^{\infty}x^{a-1}e^{-x}dx$$ then ...
0
votes
1answer
22 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
0
votes
1answer
29 views

Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
5
votes
1answer
102 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
7
votes
4answers
112 views

How to find $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$

$$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$$ Try 1: Put $z= \ln(x+\sqrt{1+x^2})$, $\mathrm dz=1/\sqrt{1+x^2}\mathrm dx$ $$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb ...
2
votes
1answer
37 views

Is $\frac{\mathrm d}{\mathrm dx} \sin x/x = \cos x/x - \sin x/x^2$ Lebesgue integrable?

Is $$\frac{\mathrm d}{\mathrm dx} \frac{\sin x}{x} = \frac{\cos x}{x} - \frac{\sin x}{x^2}$$ Lebesgue integrable? In other words, is $$ \int_{\mathbb{R}} \left| \frac{\mathrm d}{\mathrm dx} ...
4
votes
3answers
636 views

What does it mean when an integral cannot be solved in terms of elementary functions?

My calculus teacher told our class that in integral calculus, they teach you how to integrate all kinds of functions by various methods, but in the end tell you that there are infinitely many ...