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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5
votes
2answers
103 views

Volume of $ \Gamma=\{(x,y,z)\in \mathbb{R}^3:\sqrt{x^2+y^2}\le z\le \sqrt{2x^2+2y^2};x^2+y^2+z^2\le 3\} $

I evaluated the volume of the set $$ \Gamma=\{(x,y,z)\in \mathbb{R}^3:\sqrt{x^2+y^2}\le z\le \sqrt{2x^2+2y^2};x^2+y^2+z^2\le 3\} $$ by using the Pappu's centroid theorem, but I'm in trouble while ...
2
votes
0answers
68 views

Integrate $\int_0^T \frac{e^s (1+\beta e^{\theta s})}{\sqrt{1+e^s (1+\beta e^{\theta s})} }\, ds$

Integrate $$\int_0^T \frac{e^s (1+\beta e^{\theta s})}{\sqrt{1+e^s (1+\beta e^{\theta s})} }\, ds$$ I have tried integration by parts and change of variable but I'm not able to solve thus far. From ...
0
votes
2answers
37 views

Find the volume of the solid formed by the revolving the region around a line

I have to find the volume of the solid formed by the revolving the region enclosed by $x=\frac{y^2}4$ and $y=x^5+x^3$ around the line $y=2$ I know how to find the volume when it revolve around x-axis,...
3
votes
2answers
160 views

Convergence of Fourier series at $x=0$

Let $f$, $2\pi$-periodic and intergrable function defined as follows: $$f(x) = \begin{cases} 1+\sin\frac{\pi^2}{x} & x\in[-\pi,\pi),x\ne 0 \\ 1 & x=0 \end{cases} $$ Does the Fourier ...
5
votes
2answers
108 views

Improper integral of $\log x \operatorname{sech} x$

How to prove the following? $$ \int_0^\infty \log x \operatorname{sech}x\,dx = \frac{\pi}{2} \log\left( \frac{4\pi^3}{\Gamma(1/4)^4} \right) $$ I obtained the right side with CAS. It seems like this ...
4
votes
1answer
87 views

Lebesgue integrable and improper Riemann integrable

I'm sitting here on a task, where I have to show that for the function: $$f:(0,1]\times(0,1]\to\mathbb R, \quad f(x,y)=\frac{x-y}{(x+y)^3},$$ the double Riemann-integral is the double Lebesgue-...
0
votes
0answers
27 views

Change of variables for the operator curl

I am working on a finite element problem and I have some difficulties. Until now I have worked with gradients and I know how to work with this operator. But now I have to work with the curl operator ...
1
vote
0answers
62 views

Compute $\int_n^\infty \frac{\text{d}x}{x^n+\sin(nx)}$

$$\int_n^\infty \frac{\text{d}x}{x^n+\sin(nx)}$$ (In terms of $n$) I've struggled on this for quite some time because it seems to prove impossible to break up that extremely annoying $\sin (nx)$ term....
1
vote
0answers
62 views

Integrals using substitution $x\mapsto x+\frac{1}{x}$

Do you have some links to questions with integrals that require the substitution $x\mapsto x+\frac{1}{x}$ (as a trick)? I know this trick, but I can't seem to find any integrals to practice it..
1
vote
0answers
20 views

Integral over $[0,1]$ of $g\left(x^{1/(2n+1)}\right)$ vanishes

Suppose $g:[0,1] \to \mathbb{R}$ is continuous and such that $$\int_0^1 g\left(x^{\frac{1}{2n+1}}\right)\,dx = 0, \quad n=0, 1, 2, \dots$$ Show that $g \equiv 0$ on $[0,1]$. My idea was to use ...
0
votes
1answer
31 views

Evaluate $\int_{C}\frac{e^{3z}}{(z+1)^5}dz$

$$\int_{C}\frac{e^{3z}}{(z+2)^5}dz\,\,\,\,\,\,\,\,C:|z|=3$$ My try: Applaying Cauchy's formula $$\int_{C}\frac{e^{3z}}{(z+2)^5}dz=2\pi ie^{-2\cdot 3}=\frac{2\pi i}{e^{6}}$$ I'm not sure in my ...
16
votes
2answers
407 views

How to prove that $\int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}$?

A friend gave me this integral as a challenge $$ \int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}. $$ This integral can be ...
1
vote
0answers
30 views

Evaluate $\int_{C}\frac{\sin\pi(z+1)+\cos\pi z}{(z-1)(z-2)}dz$

$$\int_{C}\frac{\sin\pi(z+1)+\cos\pi z}{(z-1)(z-2)}dz\,\,\,\,\,\,\,\,C:|z|=4$$ My try: Applying Cauchy's Integral formula $$2\pi i\bigg[\frac{\sin\pi(z+1)+\cos\pi z}{z-1}\bigg]_{z=2}+2\pi i\bigg[\...
4
votes
2answers
165 views

Prove that if $f$ is Riemann integrable on $[a,b]$ then so is $f^2$

Prove that if $f$ is Riemann integrable on $[a,b]$ then so is $f^2$. I have already proved that a function is Riemann integrable if and only if it is bounded and continuous a.e. If $f$ is bounded, ...
3
votes
1answer
43 views

How can I integrate rational functions with denominator with just quadratics as factors?

I am told about integration by partial fraction method.I usually guess to decompose a fraction into partial fractions and then I solve the constants.In this problem: $$\int\frac{x^2+2x+6}{(x^2+5x+7)(x^...
4
votes
0answers
296 views

Evaluating $\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x$

I need to evaluate $$\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x $$ I've been told that the way forward is ...
1
vote
1answer
46 views

Converting cartesian double integral into polar integral

Write down the integral $$\int_0^4 \int_x^ {4x} \sqrt{x^2+y^2}dydx$$ $(i)$ integration with respect to $\theta$ first and $r$ second.
2
votes
2answers
57 views

Find the volume between two paraboloids

Find the volume of the solid enclosed by the paraboloids $z = 1-x^2-y^2$ and $z = -1 + (x-1)^2 + y^2$. Using triple integrals, it is known that $V = \iiint_R \mathrm dx\,\mathrm dy\,\mathrm dz$, and ...
3
votes
1answer
101 views

Inverse short time Fourier transform

The short time Fourier transform $S: L^2(\mathbb{R})^2 \rightarrow L^2(\mathbb{R}^2)$ can be defined as $$S(g,f)(a,b):=\int_{\mathbb{R}}f(x) \overline{g(x-a)} e^{-i b x} dx.$$ Now a natural question ...
1
vote
1answer
45 views

Integration for radial equations: $\int \limits _0 ^r \big( 2u-Au^2 \big) e^{-Au} du$

I have a Physics problem however I have an issue with an integration midway through my question. I have $$\int \limits _0 ^r \big( 2u-Au^2 \big) e^{-Au} du .$$ Where A is a constant I have tried ...
0
votes
1answer
73 views

Let $f(x)=x$ for x irrational and $f(x)=0$ for x rational (Darboux Integration)

Let $f(x)=x$ for $x$ irrational and $f(x)=0$ for $x$ rational. Show that $f$ Darboux integrable (lower and upper) on $[0,1]$ and $$(\underline{D})\int_{0}^{1}f=0,\quad (\overline{D})\int_{0}^{1}f=\...
1
vote
3answers
111 views

Elliptic Integrals

In my homework I had to solve the following integral $\displaystyle\int_0^\pi \mathrm{d}\Psi \frac{\cos\Psi}{\sqrt{1+2s(1-\cos\Psi)}}$ with some constant $s\ll1$ The solution said this is an "...
0
votes
0answers
16 views

find a solution for an integration in spherical coordinates

This problem comes from computation of magnetic field. $\vec r'$ and $\vec r$ are vectors, and represent different variable respectively. The integration is for $r'$ in Volune $V$'. Thank you!
0
votes
3answers
39 views

Integrate $dx$ over interval $a\le x \le b$ instead of just $b-a$

In the Wikipedia article on the wave function it's stated that the probability of a spin-less particle in 1D space being found in the interval $a\le x \le b$ at time $t$, where $x$ is the position, ...
1
vote
2answers
108 views

Integrating over a tetrahedron

Let $S$ be the tetrahedron in $\mathbb{R}^3$ having vertices $(0,0,0), (0,1,2), (1,2,3), (-1,1,1)$. Calculate $\int_S f$ where $f(x,y,z) = x + 2y - z$. Before I show you guys what I have tried, ...
1
vote
3answers
71 views

Integrating $\int \frac{8 dx}{3 \cos 2x + 1}$ to arctan rather than log

[Edit: It is becoming increasingly likely that the expected answer containing arctan might be a typo from my book, which is transcribed correctly here, so an answer containing log might be correct ...
4
votes
2answers
126 views

Evaluating a certain integral which generalizes the ${_3F_2}$ hypergeometric function

Euler gave the following well-known integral representations for the Gauss hypergeometric function ${_2F_1}$ and the generalized hypergeometric function ${_3F_2}$: for $0<\Re{\left(\beta\right)}<...
8
votes
2answers
390 views

Is it 'more rigorous' to perform definite integrations, rather than indefinite integration while solving ODEs?

In the beginning of my ODE course, my professor said something about performing definite integrations being 'more rigorous' than indefinite ones somehow, but also that it really wasn't much important, ...
0
votes
1answer
34 views

Area below x axis not considered when finding volume of shape rotated around y axis?

While practicing finding the volume of shapes bounded by a function and the $x$-axis by rotating the shape around the $y$-axis and taking an integral (the "washer" or "pipe" method) I became confused ...
1
vote
1answer
70 views

Changing the order of integration using polar coordinates

Write down the integral $$\int_0^1 \int_x^ {\sqrt 3x} f(\sqrt{x^2+y^2})dydx$$ as an integral/integrals in polar coordinates in two ways: $(i)$ integration with respect to $r$ first and $\theta$ ...
8
votes
1answer
68 views

Almost-identity: $[\int_0^\infty{\rm d}x-\sum_{x=1}^\infty] \prod_{k=0}^N\text{sinc}\left(\frac{x}{2k+1}\right) = \frac{1}{2}$

Show that the identity $$\int_0^\infty \prod_{k=0}^N \text{sinc}\left(\frac{x}{2k+1}\right)\,{\rm d}x - \sum_{n=1}^\infty \prod_{k=0}^N \text{sinc}\left(\frac{n}{2k+1}\right) = \frac{1}{2}$$ where $\...
1
vote
1answer
62 views

Show $\lim\limits_{n\to\infty}\left[\frac{1}{n}\sum_{i=1}^{n}f\left(\frac{i}{n}\right)\right]=\int_{0}^{1}f(x)\,{\rm d}x$

Suppose that the function $f:[0,1]\rightarrow\mathbb{R}$ is integrable. Prove that $$\lim\limits_{n\rightarrow\infty}\frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\...
5
votes
2answers
112 views

How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?

While reading physics papers I found a very interesting integral so I decided to write it down. Let $p(z) = z^ 3 - 3\Lambda^ 2 z$ where $\Lambda$ could be any number. If you want $\Lambda = 1$ and $...
1
vote
2answers
72 views

Differentiating this integral,

I want to show that $u_{xx} + u_{yy} = 0$ for the integral given below, so I think I want to differentiate under the integral with respect to both $x$ and $y$. The goal is to show that $u$ is ...
0
votes
0answers
21 views

Convergence of integrals if curve parametrisations converge

Let $\boldsymbol{r}:[a,b]\to\mathbb{R}^3$ be the piecewise continuously differentiable parametrisation of a piecewise smooth curve. If, for all $n\in\mathbb{N}$, $\boldsymbol{r}_n:[a,b]\to\mathbb{R}^...
0
votes
1answer
41 views

Differential Equations (Proving)

This is the question --> What I have done; $$ dS/dt = kS(N-S) $$ $$ 1/(S(N-S)) = k dt $$ $$ 1/(SN - S^2) = k dt $$ Therefore $$ 1/(S-2S) * ln|SN-S^2| = kt + c $$ A I on the right track? ...
1
vote
1answer
70 views

The integral in my book is wrong, can you confirm?

$$\int\sqrt{4z^2 - 4z + 2} dz$$ My book solution: $$ \left( z-1/2 \right) \sqrt{z^2-z+1/2} + 1/4 \ln\left[ z - 1/2 +\sqrt{z^2-z+1/2} \right] $$ My solution: $$ \left( z-1/2 \right) \sqrt{z^2-z+1/2}...
1
vote
4answers
57 views

Find $n$ if the area between the curve of $y=x^n$ and the $y$ axis is $3$ times the area between the curve and the $x$-axis

For this question i tried to find the area for red and blue sections, and equate them by red= 3 blue. However, it didnt work out and I got $b-a = 3(b^n - a^n)$ for the outcome. In the book, it ...
3
votes
2answers
56 views

Flux integral with vector field in spherical coordinates

I have a vector field $\vec{A}$ that is given in spherical coordinates. $$\vec{A}=\frac{1}{r^2}\hat{e}_{r}$$ I need to calculate the flux integral over a unit sphere in origo (radius 1). I cannot use ...
0
votes
1answer
109 views

Integral of a small function

How might I go about calculating $$\int_\pi^\infty \frac{\sin x}{x\log x}\text{d}x$$ I honestly don't know where to start.
0
votes
1answer
20 views

What are the limits for this triple integral?

This is probably a very easy/silly question, but still I'm not sure about it. I want to calculate the volume of a body bound between the graph of $x^2+y^2-z^4=1$ (what does it look like?) and the ...
3
votes
3answers
118 views

Evaluate $\int \frac{1}{\sin x+\sec x}\,dx $

Evaluate $$\int \frac{1}{\sin x+\sec x}\,dx $$ Expressing $\sin x$ and $\cos x$ in terms of $\tan\frac{x}{2}$ i.e. putting $\sin x=\dfrac{2t}{1+t^2}$, $\cos x=\dfrac{1-t^2}{1+t^2}$ and hence $dx=\...
4
votes
6answers
113 views

Integrate: $\int^1_0\frac{r^3}{\sqrt{4+r^2}}dr$

$$\int_0^1\frac{r^3}{\sqrt{4+r^2}}\ \mathrm dr$$ I have attached my work. I am stuck.
2
votes
1answer
131 views

Expectation defined as Riemann integral

I have a question related to the expectation of a continuous random variable and its Riemann integral definition. Consider a continuous real-valued random variable $X$ defined on the probability space ...
0
votes
1answer
53 views

How to calculate$\int_0^\pi e^{-i c (\sin(t) + \cos(t))} \sin(t)\, dt$?

I would like to calculate the following integral $$I=\int_{0}^{\pi} e^{-i c (\sin(t) + \cos(t))} \sin(t) \,dt $$ Here's what I did: We make the change of variables $s=\cos(t)$, so $$I=\int_{-1}^{1} ...
1
vote
2answers
46 views

How do I calculate $\int_0^ax^2\sqrt{a^2-x^2}dx$ via substitution?

I have to calculate the integral $$\int_0^ax^2\sqrt{a^2-x^2}dx$$ using solely substitution (no integration by parts). $a$ is a positive constant. I'm confused on how to do this?
2
votes
1answer
61 views

complex integration using residue theorem

I have the following integration $$g_n=\frac{(-1)^n}{2\pi} \int_{-\infty} ^ {\infty} (\frac {e^{-itx}}{(it-1)^n})dt $$. I used the residue theorem to find the integration so the result of the ...
2
votes
1answer
46 views

Solve the double integral $\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)dxdy\:$ [closed]

$$\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy.$$ I think you need to be solved by the transition to polar coordinates: \begin{cases} x=r\cos(\phi),\\ y=r\sin(\...
4
votes
1answer
58 views

Integration exercise #1

Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a continuous function such that $f(x)\geq 0$ for all $x$ and \begin{align*} \int_{-\infty}^{\infty}f(x) = 1 \end{align*} For $r\geq 0$, let \begin{align*} I_{...
0
votes
2answers
83 views

Integrate the square root of tangent [duplicate]

Find $\displaystyle \int \sqrt{\tan(x)}dx$. According to an integral calculator, the answer to this question is $$(-2 \tan^{-1}(1-\sqrt(2) \sqrt(\tan(x)))+2 \tan^{-1}(\sqrt(2) \sqrt(\tan(x))+1)+\log(...