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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16
votes
3answers
185 views

Finding integral of a function

I have stumbled upon an exercise that reads thus: $$\int\limits_{-\infty}^0\frac{x^x}{x^3-1}\mathrm{d}x=\frac{2\sqrt3}{9}\pi,$$ and I am guessing it is asking to prove the above equality. ...
13
votes
2answers
487 views

Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$

I am asking this question out of curiosity. $$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha ...
0
votes
0answers
23 views

Finding the integral of arccos(x) to be 0.5 [closed]

The integral from k to 0 of arccos(x) with respect to x is 0.5. Ok, so I came as far as: karccos(k) - sqrt(1-k^2) = -0.5 But from here, if the right path, I failed to find the solution. Any help ...
1
vote
1answer
50 views

Autonomous differential equation

Let $f: \Bbb R \to \Bbb R$ and $x_0 \in \Bbb R$, such that $f(x_0)> 0 $, and assume that $x(t)$ is the solution of $x'=f(x)$, such that $x(0)=x_0$. If $f(x) > 0$ then $x(t)$ is defined for all ...
1
vote
1answer
23 views

solving this integral

$$\int_0^1 \int_0^\eta \phi \, d\eta' d\eta$$ $$\phi=\eta-3\eta^3+2\eta^4$$ This is from a fluid mechnanics paper. Can someone solve it and tell me what dn' means since n is a variable already? ...
0
votes
1answer
42 views

Use complex number to solve this equation $\int e ^{3x} cos x dx$?

I can solve it another way, but am not sure how to use complex numbers to solve it. Thanks for your help
-2
votes
2answers
22 views

Can someone ple help me answer ths questions. Calculus || [closed]

Consider curves: f(x) = x^3 and g(x) = x^2 (a) Find all points of intersection of these two curves. (b) Find the area between these two curves.
0
votes
1answer
24 views

Finding Volume with Solids of Revolution [closed]

Find the volume of the solid obtained by revolving $y=8-x^3$ about the $x$-axis from $x = 0$ to $x = 2.$
0
votes
0answers
18 views

Checking some work on an expectation value problem

I am working on a pretty simple problem (or so it seems it should be) from Griffith's QM text. The problem states: for the probability density function $\rho (x) = Ae^{-\lambda(x-a)^2}$ a) find A ...
4
votes
1answer
51 views

integral involving hypergeometric function $\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{y})}{y}\,dy$

I arrived at the following result $$\tag{1}\int^\infty_0 z^{p-1} E^2(z)\,dz=\frac{\Gamma(p)}{p}\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{z})}{z}\,dz$$ where the exponential integral $E(z)$ is defined ...
-1
votes
2answers
45 views

Simple integration question y prime and y^2 [closed]

How does one integrate $y'/y^2$ in terms of y? I'm stumped.
0
votes
2answers
41 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
0
votes
1answer
17 views

Finding volume of solid using two methods

The question is: The region bounded by $y=\frac{1}{x}, y=0, x=1, x=2$ is rotated about the $y$-axis, thus creating a solid. Compute the volume using the Shell and Slicing method. This is what I have ...
8
votes
1answer
129 views

Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$

One of the possible ways of computing the series is to obtain the generating function, but this might be a tedious, hard work, pretty hard to obtain. What would you propose then? ...
0
votes
0answers
22 views

Comparing values of integral after and before applying a diffeomorphim

Let $\phi:\mathbb{S}^1\to\mathbb{S}^1$ be a diffeomorphim such that its Jacobian satisfies $\int_{\mathbb{S}^1}\operatorname{Jacobian}(\phi)=2\pi$, and let $f:\mathbb{S}^1\to\mathbb{S}^1$ be a smooth ...
0
votes
1answer
56 views

Prove: If $f,g$ agree on $A\subseteq [a,b]$ then $\int_a^b f = \int_a^b g$.

Prove: If $f,g$ agree on $A\subseteq [a,b]$ then $\int_a^b f = \int_a^b g$. In short, the proof starts with: We'll choose partitions, $\Pi_n$ such that $\Pi_n \to \infty$. And they key in this ...
0
votes
0answers
46 views

Integral Problem

A particle of mass $m$ is projected towards a point $O$ with initial speed $\dfrac{\sqrt{5}}{3}$ m/s from a point $P$, where $\overline{OP}$ is 3 meters. The particle is repelled from $O$ by a force ...
1
vote
1answer
28 views

Lesbegue integration w.r.t scaled measure

I was wondering if we have a measure $\mu$ and $a \in \mathbb{R}$ with $\lambda = a\mu$ if we get : \begin{align} \int fd\lambda = a\int fd\mu \end{align} It seems to work for the basic definition of ...
1
vote
1answer
32 views

Different answer when simplifying before integrating

I have been trying to get my head around this for some time now... I solve the same integral in two ways but get two different solutions. Since there can't (surely) be any sort of ambiguity when ...
1
vote
2answers
58 views

Integral involving exponents

How do we integrate $\int e^{C_1\frac{u^2+1}{u^2-1}} \ du\tag 1$ I could not find a proper substitution to convert it to a normal available form so that I can get a closed form of integration. $C_1$ ...
2
votes
1answer
276 views

Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where ...
3
votes
1answer
53 views

Stuck on tough integral

I am trying to solve what looked like a simple integral but I got a bit stuck. The integral is : \begin{equation} \int_0^x \frac{ab(1-e^{-ct})}{d-\frac{b(1-e^{-ct})}{c}}dt \end{equation} I tried ...
0
votes
2answers
50 views

Is there an easier way to integrate $\int \frac{d\, x}{\sin^2{x}}$?

So, hey. I've made it like this: $ \tan{\frac{x}{2}} = t, x=2\arctan{t}, d\, x =\frac{2t\,d\,t}{1+t^2}$, therefore $\int\frac{d\, x}{\sin^2{x}} = \int\frac{1+t^2}{2t^2}d\,t$, which gives us ...
0
votes
2answers
47 views

Integration by substitution: What formula can I refer to?

When I am trying to integrate a composite function $f(g(x))$ that is multiplied by $g'(x)$, then there's a formula for that in my book. It's simply $F(g(x)) + C$. But what if $g'(x)$ isn't there? ...
0
votes
1answer
81 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
0
votes
0answers
22 views

Exchanging the limit of an integral with a finite sum

So, in general, I can get this value: $$\lim_{a \to \text{a constant}}{ \int{ \left( \sum_{x=x_1}^{x_2}{ f(a,x) }da \right)} } \tag{1}$$ What I'm after is this: $$\sum_{x=x_1}^{x_2}{\left( \lim_{a ...
4
votes
1answer
179 views

Calculating $\int \dfrac{\cot^3x}{\sqrt{1+\csc^4x}}\;dx$

Evaluate:$$\int \dfrac{\cot^3x}{\sqrt{1+\csc^4x}}\;dx$$
48
votes
6answers
2k views

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
6
votes
2answers
71 views

A pseudometric on the space of the measurable functions is complete

I'm working in the following exercise: Suppose $(X, \mathcal A, \mu)$ is a finite measure space and suppose $\mathcal F$ is the set of all $\mathcal A$-measurable functions $f: X \rightarrow \mathbb ...
0
votes
0answers
63 views

Is there a 4 pointed star that is regular?

I am studying about the area of a 4 pointed star, I wonder if there is really a 4 pointed star that is regular? what could be the characteristics of a regular 4 pointed star? that is how i ...
2
votes
2answers
71 views

how to integrate $\sqrt{1-x^{2/3}}$

i'm facing the following problem: find the area of the set $M=\{(x,y): |x|^{\frac{2}{3}}+|y|^{\frac{2}{3}}\le 1\}$ using integration i thought about only integrate where x,y both $\ge0$ and multiply ...
1
vote
2answers
67 views

Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
7
votes
1answer
112 views

Compute polylog of order 3 at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order 3 at $\frac{1}{2}$, but how to prove it using integral or Euler sum only (without ...
0
votes
2answers
42 views

Prove that $\displaystyle\int_0^\pi \frac{dx}{a^2 \cos^2 x+b^2 \sin^2 x} = \frac{\pi}{ab}$ [closed]

I need solution for Prove that $\displaystyle\int_0^\pi \frac{dx}{a^2 \cos^2 x+b^2 \sin^2 x} = \frac{\pi}{ab}$ Help me to find it as early as possible
0
votes
1answer
25 views

How I integrate this function? with Delta Fuction

here is $$∫_0^l F(x,ζ)ϕ(x)dxdζ$$ with this $$F(x,ζ)=Q(ζ)δ(x-x0) e^{-rx}$$ Thanks for your help!
2
votes
1answer
50 views

A proof involving nested integrals and induction [duplicate]

Prove that $$\int_0^x dx_1 \int_0^{x_1}dx_2 \cdots \int_0^{x_{n-1}}f(x_n) \, dx_n =\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t) \, dt$$ I'm trying induction over $n$. The case $n=1$ is trivial. When ...
1
vote
1answer
43 views

invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of "Homotopy invariance of line integral ...
-1
votes
1answer
52 views

Is the piecewise function that is equal to $1/x$ on $(0,1]$ Riemann integrable?

Is this function Riemann integrable? $$f(x)=\begin{cases}0\quad &\text{ if }x=0 \\ 1/x \quad &\text{ if }0 < x \leq 1\end{cases}$$ I think it is. $0$ is just one single point, which won't ...
3
votes
2answers
76 views

$\int\frac{dx}{x-3y}$ when $y(x-y)^2=x$?

If y is a function of x such that $y(x-y)^2=x$ Statement-I: $$\int\frac{dx}{x-3y}=\frac12\log[(x-y)^2-1]$$ Because Statement-II: $$\int\frac{dx}{x-3y}=\log(x-3y)+c$$ Question: Is ...
9
votes
2answers
223 views

Closed form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
6
votes
3answers
458 views

Prove that $\,f=0$ almost everywhere.

Let $f$ be a Lebesgue integrable function on $[0,1]$ such that, for any $0 \leq a < b \leq 1$, $$\biggl|\int^b_a f(x)\,dx\,\biggr| \leq (b-a)^2\,.$$ Prove that $f=0$ almost everywhere. I would ...
6
votes
1answer
164 views

Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{\operatorname dx}{\operatorname \Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of ...
0
votes
3answers
51 views

How to find the integral of $(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}$

I have a function that need calculate the integral. Could you help me to find it. Thank you so much $$f(\omega)=\int_{-1}^1(1-|\tau|)\cos(\omega\tau)e^{-j\omega\tau}d\tau$$ where $\omega$ is constant. ...
7
votes
4answers
3k views

Help finding integral: $\int \frac{dx}{x\sqrt{1 + x + x^2}}$

Could someone help me with finding this integral $$\int \frac{dx}{x\sqrt{1 + x + x^2}}$$ or give a hint on how to solve it. Thanks in advance
2
votes
3answers
86 views

How to find the antiderivative of this function?

I want to integrate this function: $$\int\dfrac{x^2}{e^x-1}dx$$ I used integration by parts formula to integrate it. However I have reached somewhere where I got something like this: ...
0
votes
2answers
50 views

Integrate $\int\frac{dx}{x\sqrt{x^2+x+1}}$ [closed]

Hello I need some help with the following integral: $$\int\frac{dx}{x\sqrt{x^2+x+1}}$$ Have been trying u-sub, and parts which do not get me to a solution!
9
votes
1answer
107 views

Integral of $\sqrt{x^3 + 8}$?

I have issues solving the following integral: $$\int\sqrt{x^3+8}~dx$$ I tried substitution and integration by parts, but with no use. I'm guessing I have to use some trigonometric substitution. ...
1
vote
2answers
50 views

Double integral of $\dfrac{y}{x^2y^2+1}dx~dy$

I'm trying to solve the double integral $\displaystyle\int_0^1\int_0^1\dfrac{y}{x^2y^2+1}dx~dy$ . I'm guessing something with natural log will have to be done. Doing the steps of this problem are more ...
30
votes
3answers
3k views

How do people apply the Lebesgue integration theory?

This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas. In a lot of calculus textbooks, there is ...
1
vote
0answers
31 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...