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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8 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?
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1answer
67 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 ...
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0answers
36 views

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

$$\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 ...
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0answers
37 views

How to calculate the area of a 4 dimensional curve?

I have been searching on Google about it, and I found that given a sample, 4 points, per example, I could find a function and use integral on it. I am sorry if it sounds silly, I am very dummy in ...
2
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3answers
76 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
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1answer
56 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 ...
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0answers
44 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 ...
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1answer
32 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 ...
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2answers
104 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 ...
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4answers
57 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 } ...
5
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0answers
46 views

Integral of $\int \frac{x^2-4}{x^2\sqrt{4+x^2+x^4}} \,dx$

$$\int \frac{(x^2-4)dx}{(x^2\sqrt{4+x^2+x^4})}$$ My try: $$\int \frac{(1-4/x^2)dx}{(\sqrt{4+x^2+x^4})}\\ =\int \frac{(1-4/x^2)dx}{(\sqrt{(x^2+1/2)^2+15/4})}\\$$ Let $t=x^2$ $$=\int ...
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0answers
11 views

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

How Can I solve $F(x)=(x+y)e^{x-y}$, Find $\dfrac{\partial F}{\partial x}$ and $\dfrac{\partial F}{\partial y}$
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1answer
32 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| - ...
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0answers
28 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
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3answers
142 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 ...
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2answers
16 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 ...
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1answer
25 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 :: ...
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3answers
51 views

For what $p$ does this series converge

"Find the values of $p$ s.t. the following series converges: $\sum_{n=2}^{\infty} \frac{1}{n^p \ln(n)}$" I am trying to do this problem through using the Integral Test to find the values of $p$. I ...
13
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4answers
177 views

How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$$
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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} ...
1
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1answer
71 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.
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1answer
73 views

Integrating $\int\frac{x^2+1}{(x-1)^3(x+2)}\mathrm dx$ [on hold]

I am struggling with the following integral: $$\int\frac{x^2+1}{(x-1)^3(x+2)}\mathrm dx$$ I guess it all comes down to a fairly simple algebraic manipulation - but I cannot see it...
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0answers
39 views

Integral of Normal Distribution with imaginary unit

Hi I need some help with the following integral. $$ \int_{-\infty}^{\infty} \operatorname{e}^{itx} \cdot \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \operatorname{e}^{\frac{-(x - \mu)^2}{2\sigma^2}} \mathrm ...
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1answer
28 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
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0answers
28 views

Volume and surface area of elliptic torus

ok, so i have an elliptic torus, parametrised as (i'll just copy the mathematica syntax): rr := {(R + a Cos[v]) Cos[u], (R + a Cos[v]) Sin[u], b Sin[v]} R is here the 'big' radius, and a and b are the ...
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1answer
30 views

Explanation of the passage from $\int_{N'}^N dN/N$ to $\ln N-\ln N'$

While going through my text I got stuck in the derivation given in the picture. ($\Omega$ is a constant) I don't know how to get the second step from the first step, also I don't know why ln is ...
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1answer
49 views

Proving that $\displaystyle \int_{0}^{a} f(x) \;\mathrm dx = \int_{0}^{a} f(a - x) \;\mathrm dx$

The question I have is: Prove that $\displaystyle \int_{0}^{a} f(x) \; \mathrm dx = \int_{0}^{a} f(a - x) \; \mathrm dx$ Since this question occurs at the end of an exercise on integration by ...
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0answers
54 views

Equality case in Hölder's inequality

How can I show that $$\left(\int{p(x)^{1-\sigma}\mathrm dx}\right)^{\frac{1}{1-\sigma}}\cdot \left(\int y(x)^\frac{\sigma-1}{\sigma}\mathrm dx\right)^{\frac{\sigma}{\sigma-1}}=\int p(x) ...
2
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1answer
73 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 ...
0
votes
1answer
33 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
3
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1answer
68 views

How to find $\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\mathrm dx$

I have a integral which seems difficult to me. Any help would be appreciated. Find $$\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\mathrm dx$$ Also I wound like to know ...
2
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0answers
38 views

Fitzpatrick's proof of Darboux sum comparison lemma

I am just reading Fitzpatrick's advanced calculus. He wants to prove $\lim (\max(x_{i-1} - x_i)) =0$ and $\lim(U(f,P)-L(f,P))$ is equivalent to $f$ is integrable. He used darboux sum comparison ...
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0answers
24 views

How should I use the integral in this problem?

Let's say there is a charged tube(cylinder with no top or bottom) with radius $a$, length $l$ and charge $q$ and a point which is collinear with the centre of the charged tube. Anyway, since we can ...
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0answers
42 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
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2answers
68 views

Integral of inverse of square root of a quadratic

I haven't taken a course on calculus so far so I don't know what to do. The integral may be wrong. Please tell me which part of it is wrong. $$ q∫_{+a}^{-a}\lim_{c \to g}\frac 1{(b^2+c^2)^{3⁄2}} dc $$ ...
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0answers
13 views

how to increace the volume to a specific volume in revolution of solid, using integration [on hold]

two functions, f(x)= 1/9(x-2)^2+7 domain range:{0,10}, g(x)=1/7(x-5)+0.7 domain range: {10,13} increase the volume to 1000ml to 1050mL using integration.
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4answers
73 views

Does $\int_{-\infty}^\infty \frac{\mathrm dx}{(1+x^2)^\alpha}$ converge?

I'm wondering when the integral $$ \int_{-\infty}^\infty \frac{\mathrm dx}{(1+x^2)^\alpha} $$ converges for the real number $\alpha$.
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4answers
64 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
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1answer
58 views

Evaluate the integral with respect to x. [on hold]

I don't know how to begin. Any tips and the correct answer would be appreciated. Evaluate the integral $$\int4^{(x+4^x)}dx$$ Sorry, I read the original problem incorrectly.
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votes
2answers
87 views

Evaluate $\int\frac{8x+20}{5x^2+25x+20}dx$

I tried to solve it and got $\frac{4}{5} \ln(4+5 x+x^2)+C$ as an answer, but my online homework program says it's incorrect. What did I do wrong? I pulled out $\frac{4}{5}$ as a constant and saw ...
2
votes
3answers
53 views

Calculate the area of the ellipsoid that rotates around the $x$-axis

So we are about to calculate the area of the ellipsoid around the $x$-axis. $$ \frac {x^{2}}{2}+y^{2} = 1 \implies x=\sqrt{2-y^{2}}$$ We are squaring it so the sign shouldn't matter. I was ...
3
votes
3answers
63 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
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1answer
31 views

Solid Angle Integration

Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a ...
1
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0answers
26 views

Formally evaluating integral to calculate electric or gravitational field.

I never understood how such integrals are calculated, formally. In a line is easy, just a line integral. In a surface, sometimes is easy, like in a disc. But, some surfaces, like sphere, it gets ...
1
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1answer
36 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
2
votes
6answers
129 views

Proving that $\int \frac{1}{x} \mathrm dx = \ln(|x|) + C_1$

In all textbooks and online notes, there is always a table of antiderivatives and it always says $\int \frac {1}{x} \mathrm dx = \ln(|x|)+C_1$ but there is nowhere a proof. I found some proofs online ...
1
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1answer
60 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
6
votes
2answers
98 views

improper integral containing $\sqrt{\cos x-\dfrac{1}{\sqrt 2}}$ in the denominator

How do i find the value of this integral-- $$I=\displaystyle\int_{0}^{\pi/4} \frac{\sec^2 x \ dx}{\sqrt {\cos x-\dfrac{1}{\sqrt 2}}}$$ I came across this integral too in physics.
7
votes
4answers
99 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 ...
0
votes
1answer
41 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...