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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0
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1answer
198 views

Shell method around a vertical line

I've been reviewing the shell method for my Calculus II final, but I suppose I need a little refresher. For example, I am given the equations $y=2x+15$ and $y=x^2$ and told to revolve the enclosed ...
10
votes
2answers
810 views

How can a function have an antiderivative that can't be written?

If a function's integral can't be written, then how can we find exact values for it over areas? Can we only ever estimate it? Why can't we make new functions to define these strange unwritable ...
2
votes
3answers
108 views

Evaluating $\int (\tan^3x+\tan^4x) dx $ using substitution

solving $$\int (\tan^3x+\tan^4x) dx $$ using substitution $$t = \tan x$$ My approach has led me to $ \int (1+t)t\sin^2xdt$ which has an $x$ too much and isn't easily solvable for me. If I remove the ...
2
votes
1answer
51 views

Double integral definition

I don't understand the definition of double integral. For instance in the functions with single variable the definite integral was defined as Riemannian sum as: $$\lim_{n\to\infty}\sum_{k=1}^{n} ...
3
votes
3answers
131 views

Need some advice to solve this integral $\int\frac{\sin^2x}{1+\sin^2x}\mathrm dx$

I'm trying to use this subtitution $t=\tan(x/2)$. But I don´t get anywhere. I've tried $t=\tan(x)$ too. Appreciate your help. $$\int\dfrac{\sin^2x}{1+\sin^2x}\mathrm dx$$
1
vote
1answer
50 views

Confusing Triple Integral - Volume

Compute the volume of the solid bounded by the cone $z = 3\sqrt{x^2 + y^2}$, the plane $z=0$, and the cylinder $x^2 + (y-1)^2 = 1$. So I tried parametrizing with $x = r\cos\theta$, $y = r\sin\theta ...
1
vote
1answer
47 views

Computing an integral via integration by parts

Good afternoon, I am learning for Monday's test and I stuck on this example: $$ \int (2x+3) \, \cos{2x} \, \mathrm{d}x $$ and I am asked to determine this integral. I am trying to do it by ...
4
votes
0answers
116 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
10
votes
1answer
259 views

Can I flip the integral and sum here?

I have the convergent integral-sum: $$\int_0^\infty \sum_{n\mathop=0}^\infty \frac {x^{4n+1}} {e^x - 1} \frac {(-1)^n} {(2n)!(4\pi)^{2n}}\mathrm d x$$ But is it the same as this?: ...
0
votes
1answer
47 views

Computing integral with integration by parts

I am trying to figure out the example below, but I still cannot get the right result (I don't know it though, I am just sure this is not the right one). What should be the proper procedure to solve ...
0
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0answers
19 views

Obtaining $g$ an envelope function and $\int_a^b g(x)dx = 1$

Suppose that we have a function $h(x)$, which is defined on an interval $[a,b]$ and we want to obtain a function $g(x)$ from $h$ such that: 1) $\int_a^b g(x)\; dx = 1$ 2) for a given function ...
1
vote
1answer
51 views

Evaluating $\int_0^\infty \int_0^\infty \frac{x^2 + y^2}{1 + (x^2 - y^2)^2} e^{-2xy} \:\mathrm{d}x \:\mathrm{d}y$

I am trying to evaluate the following definite double integral: $$\int_0^\infty \int_0^\infty \frac{x^2 + y^2}{1 + (x^2 - y^2)^2} e^{-2xy} \:\mathrm{d}x \:\mathrm{d}y$$ I have tried the following ...
5
votes
1answer
125 views

Integral inequality $\int_0^{+\infty}|\frac{\sin x}x|^p dx\leq\frac\pi{\sqrt{2p}}$

$p\geq2$, then we have $$\int_0^{+\infty}\Bigg|\frac{\sin x}x\Bigg|^p\,\mathrm dx\leq\frac\pi{\sqrt{2p}}$$ I try to use $\Bigg|\frac{\sin x}x\Bigg|\leq1$, and $\frac{\sin ...
2
votes
1answer
28 views

Simple integration problem

I need to find the work done by some stationary particle on another. The work is defined as a scalar product of force and displacement ($W=\int_{}^{} \vec{F} \cdot d \vec{r}$). The force is inversely ...
4
votes
2answers
190 views

Definite integral $\int_{R_0}^{R}\frac{dr}{r^2\sqrt{\frac{R_0-R_S}{R_0^3}-\frac{1}{r^2}\left(1-\frac{R_{s}}{r}\right)}}$

In general relativity, null geodesics (in the unbounded case) can be written under the following form : ...
2
votes
4answers
57 views

Prove that integral is independant of its parameter

We are given intergral $\int_0^\infty {\frac {dx} {(1+x^2)(1+x^\alpha)}}$ and task is to prove that it's independent of $\alpha$. Task was too complicated for me, so I had to stick with solution, ...
0
votes
3answers
114 views

How to solve $\int{\sqrt{1-x^8}}dx$

I have tried to solve it with substitution, but I always have extra x variable that I can`t get rid off. $$\int{\sqrt{1-x^8}}dx$$ Can you help me to solve this integral, how can I start? Thanks!
3
votes
2answers
54 views

Solving $\int \frac{x\ln x}{(1+x^2)^2}$

Solving $$\int \frac{x\ln x}{(1+x^2)^2}$$ This should probably be done with substitution and partial integration. However I fail to find a good substitute. I tried $u = \ln x$ but that led me to an ...
5
votes
4answers
228 views

Integral$\int_0^{\pi/4} \log \tan \left(\frac{\pi}{4}\pm x\right)\frac{dx}{\tan 2x}=\pm\frac{\pi^2}{16}$

Hi I am trying to prove $$ \int_0^{\pi/4} \log \tan \left(\frac{\pi}{4}\pm x\right)\frac{dx}{\tan 2x}=\pm\frac{\pi^2}{16}. $$ What an amazing result and a clever one this is. I tried writing $$ ...
4
votes
1answer
130 views

Integral $\int_0^\infty \ln x\,\exp\left(-\frac{1+x^4}{2\alpha x^2}\right) \frac{x^4+3\alpha x^2- 1}{x^6}dx$

$$I:=\int_0^\infty \ln x\,\exp\left(-\frac{1+x^4}{2\alpha x^2}\right) \frac{x^4+3\alpha x^2- 1}{x^6}dx=\frac{(1+\alpha)\sqrt{2\alpha^3 \pi}}{2\sqrt[\alpha]e},\qquad \alpha>0.$$ This one looks very ...
-1
votes
2answers
59 views

Why is the integral of $\cos(x) - \sin(x)$ from $0$ to $π/4$ equal to $\sqrt2 - 1$?

Why is $$\int_ {0}^{π/4} {\cos(x)} - {\sin(x)} \ \mathrm{d}x=\sqrt2 -1$$ This answer popped up on a problem I was doing and it piqued my interest. Can anyone help me out?
6
votes
3answers
264 views

How would you evaluate $I:=\int_ {0}^{\infty} \frac {\cos(ax)} {(x^2 + b^2)^n} \ \mathrm{d}x$?

Any pointers on how should I start? $$I:=\int_ {0}^{\infty} \frac {\cos(ax)} {(x^2 + b^2)^n} \ \mathrm{d}x$$
3
votes
1answer
133 views

Need help evaluating a definite integral $\displaystyle \int^{\infty}_{-\infty} \frac{dx}{(1+4 x^2)\cosh(\pi x)} = \ln2$

Can anyone show how to evaluate this integral? $\displaystyle \int^{\infty}_{-\infty} \frac{dx}{(1+4 x^2)\cosh(\pi x)} = \ln2$
2
votes
1answer
35 views

Is the implication $f$ (improperly) Riemann integrabel then $f$ is measurable true?

The title more or less says it: Does it follow that if $f$ is Riemann integrabel, then $f$ is measurable (on arbitrary sets $A \subset \mathbb{R}^n)$? Does this theorem have a name?
2
votes
2answers
107 views

Integration problem: $\int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx$

I have to integrate $$\int _ {-\infty} ^ {\infty} \frac {e^{\large-x^2}}{\sqrt{\pi}} e^{\large x}\ dx.$$ I've already done by numerical approximations, like Simpson's rule and Gauss-Hermite, but I ...
9
votes
3answers
557 views

Odd $\sin/\cos$ integral

How to evaluate $$\int \frac{\sin^3 x}{\cos^5x}dx\ ?$$ I've tried various substitutions with $\sin x = u$ or $\cos x = u$, I've tried using Euler's formula which result in too heavy calculations and ...
1
vote
2answers
88 views

Is there any problem on evaluating the indefinite integral that couldn't be solved?

In another word, is there any indefinite integral that couldn't find another representation
1
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3answers
111 views

Integral with goniometric functions $\int(1+\cos^2x-\sin^2x)dx$

I am solving this example: Transcription: \begin{align} &\int(1+\cos^2x-\sin^2x)dx=\int(1+1-\sin^2x-\sin^2x)dx=\int(2-2\sin^2x)dx=\\ ...
0
votes
1answer
32 views

Calculating a line integral

Calculate $\int_\gamma f(t) dt$ where $f(t) = t^2$ and $\gamma $ is the semi circle from $i$ to $-i$ (counter clockwise) I set $\gamma : \left[\frac{\pi}{2},\frac{3\pi}{2}\right] \rightarrow ...
1
vote
1answer
84 views

What is the proof of integral theorem i.e area under curve is given by anti derivative?

I have learnt integration as well as differentiation. In the early days I learnt a very simple proof for why the derivative of sin(x) is cos(x) and that of tan(x^2) is 2x*sec(x^2). This basically ...
1
vote
1answer
77 views

Manipulation of Cauchy's Integral Formula

$\quad$ Using Cauchy's integral theorem, write down the value of a holomorphic function $f(z)$ where $|z|\lt1$ in terms of a contour integral around the unit circle, $\zeta=e^{i\theta}$. $\quad$ ...
3
votes
2answers
136 views

How to evaluate this integral? $\int \frac {x e^{\arctan(x)}}{{(1+x^2)}^{3/2}} \ dx$

I am working on a integral and I run out of ideas how to solve it. Does anyone has good some good idea? I tried various substitutions but it seems that I did not find the correct one. $$\int \frac ...
4
votes
1answer
76 views

Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
7
votes
4answers
340 views

Evaluate $\int\frac{1}{\sin(x-a)\sin(x-b)}\,dx$

I'm stuck in solving the integral of $\dfrac{1}{\sin(x-a)\sin(x-b)}$. I "developed" the sin at denominator and then I divided it by $\cos^2x$ obtaining ...
1
vote
2answers
77 views

Evaluate limits by interpreting sums as integral sums

Problem: Evaluate the following limits by interpreting given sums as integral sums for certain functions and by using the Fundamental Theorem of Calculus. (a) Find $\lim{S_{n}}$ as n goes to infinity ...
2
votes
0answers
51 views

Calculating this integral: $I=\int_{0}^{\infty}(\log t)\,(\tan^2t)\,\mathrm{d}t$

How to calculate this integral? $$I=\int_{X_0}^{X}(\log t)\,(\tan^2t)\,\mathrm{d}t.$$ I tried integrate by parts and I found something related to: $$J=\int_{X_0}^{X}\dfrac{\log\cos ...
0
votes
0answers
25 views

Is there any relationship between double and single integrals?

So here is my question, Let $D:=[-d,d]\times[-d,d]\subset\mathbb R^2$ for some positive $d$. Moreover assume that $f:D\rightarrow\mathbb R$ is a smooth function with compact support in $D$ and that ...
1
vote
2answers
64 views

Solving $\int_0^1 \frac{dx}{e^x-e^{-2x}+2}$ with substitution

$$\int_0^1 \frac{dx}{e^x-e^{-2x}+2}$$ I've tried solve this with substitution, first with $ u = e^x$ witch gives me $\int_0^1 \frac{1}{u\left(1 - \frac{1}{e^2}\right)+2} du$ and second with $ u = ...
1
vote
0answers
165 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
8
votes
5answers
406 views

Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
1
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0answers
24 views

Integrate a divergence-free vector field

Suppose we are given a vector field $\overrightarrow{B}$ in $\mathbb{R}^3$ whose divergence is zero : $div(\overrightarrow{B})=0$. We want to find $\overrightarrow{A}$ such that ...
0
votes
2answers
53 views

Need help finding Jacobian matrix of diffeomorphism of spheres

Let $S_a \subset \mathbb{R}^{n+1}$ and $S_b \subset \mathbb{R}^{n+1}$ be two spheres of radius $a$ and $b$ respectively. So $S_a$ are $n$-dimensional. Let $F:S_a \to S_b$ be the diffeomorphism $F(s) ...
1
vote
1answer
29 views

A Condition for the Integrand of an Absolutely Convergent Integral

Given that $f(x)$ is continuous on the Real line and that $$\int |f(x)|d\mu < \infty$$ is it a necessity that $\lim_{t \to \infty}f(t)=0$? I tried very much to get a counterexample but cannot ...
1
vote
1answer
71 views

solve $\int_0^1\cos x^{1/3} dx$

solve $$\int_0^1\cos x^{1/3} dx$$ My approach would be to use substitution and set $\cos x =t, dx = - \sin (t) dt$ which gives me $-\int_0^1t^{1/3} \sin( t) dt$ however then I'm stuck again...
2
votes
1answer
57 views

Find $f(x,y)$ such that $\int\int f(x,y)dxdy \neq \int\int f(x,y)dydx$ how can this be?

The question is: find a function $f: [0,1]^2 \to \mathbb R$ such that $\int_{0}^{1} \int_{0}^{1} f(x,y)dxdy \neq \int_{0}^{1} \int_{0}^{1} f(x,y) dydx$ and both integrals exist. I'm not saying that ...
29
votes
5answers
2k views

$\ln|x|$ vs.$\ln(x)$? When is the $\ln$ antiderivative marked as an absolute value?

One of the answers to the problems I'm doing had straight lines: $$ \ln|y^2-25|$$ versus another problem's just now: $$ \ln(1+e^r) $$ I know this is probably to do with the absolute value. ...
2
votes
0answers
23 views

Would like to calculate the following limit:

$\lim_{n \to \infty} \int_0^1 \arcsin (\sin(nx)) dx$ I think the answer is 0, but can't prove it.
0
votes
0answers
31 views

Why does the 1/2 vanish when finding this antiderivative?

Integrating: dx/sqrt(x)+2x I substituted u=sqrt(x) and got u/2u+2u^2 I pulled out u and got 1/2+2u and simplified to 1/2(1+u) but my book says the answer is ...
4
votes
4answers
179 views

Why is calculating the area under a curve required or rather what usage it would provide

I understand Integration and Differentiation and see a lot of Physics / Electrical Theory using them. Take for example a sine wave. So area for me means the space any object would occupy. So what's ...
0
votes
4answers
84 views

Integral of $16/(1-\cos8x)$

Can someone please help me with this question: $$ \int \ \frac{16}{1-\cos8x} \ \ dx \ \ . $$ I tried substitution by letting $u=1-\cos8x$, it got messy after the substitution. I used the ...