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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17
votes
10answers
2k views

What does it mean when dx is put on the start in an integral?

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
0
votes
0answers
21 views

hard integral problems to solve

I'm practicing harder integration using techniques of solving with special functions I have difficulties with these two hard integrals; don't even know how to start, please help me to start of ...
0
votes
1answer
18 views

Evaluate $\iiint_V zdV$, V is volume bounded below by cone $x^2+y^2 = z^2$ and above by sphere$ x^2+y^2+z^2=1$,lying on positive side of y-axis.

$\iiint_V zdV = \iiint_{\sqrt{x^2+y^2}}^{\sqrt{1-x^2-y^2}} zdzdydx = \iint_D \frac{1-2(x^2+y^2)}{2}dxdy$ where D is given by the disc $x^2+y^2 = \frac{1}{2}$ Changing x,y into cylindrical coordinates,...
0
votes
0answers
16 views

Integral over domain with implicit boundary

I have a unit disc (circle) $$x^2 + y^2 = 1$$ and a function $$\sum_{i=0}^N c_i\left(\sqrt{(x-a_i)^2 + (y-b_i)^2}\right)^3 = 0$$ I want to compute an area bounded by these two functions for $x$ ...
22
votes
3answers
1k views

Does integration by parts with “deja vu” have a name?

In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
4
votes
2answers
86 views

Compute $\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$

I am trying to compute the integral $$\int\limits_Q^1\sqrt{(1-x^2)(1-\frac{Q^2}{x^2})}\mathrm{d}x$$ where $0\leq Q <1$ is a real number. I tried to substitute $x=\cos y,$ but this didn't bring ...
0
votes
1answer
13 views

Infinities on null sets

This is a conceptual question! Why is it that (e.g.) $\int_0^1 \frac{1}{x} dx$ doesn't converge. I'm stuck in the following way of thinking about it: Since the problematic part is $\int_0^\epsilon \...
4
votes
2answers
84 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
0
votes
0answers
14 views

Volume of the $N$-dimensional domain $\sum\limits_{k=1}^N (1 + |x_k|^a)^b\le\epsilon$

I wish to calculate the following $N$-dimensional integral $$I = \int_0^\infty dx_1 \ldots \int_0^\infty dx_{N} \, H\left(\epsilon - \sum_{k=1}^N (1 + x_k^a)^b\right),$$ where $a, b$ and $\epsilon$ ...
0
votes
1answer
23 views

Integral substitution

I don't understand why the integral boundary change here from $[0,1]$ to $[0,\infty]$ $$\int_0^1 \int_{0}^\infty xe^{-x}f(ux,(1-u)x)\mathrm{d}u \mathrm{d}x$$ Substitution: $(ux=t,\ (1-u)x=s)\implies ...
4
votes
2answers
160 views

How to integrate $\int \dfrac{x^{13}\ dx}{x^5 + 1}$

We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method: $$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$ I cant find way to integrate ...
2
votes
1answer
20 views

Converting Ellipse Integration Boundaries To Cylindrical Coordinates

I'm having the following integral, and I'm being asked to convert the integration boundaries to cylindrical coordinates. I've figured out that on XY-plane it's an ellipse having the following ...
11
votes
3answers
454 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\...
1
vote
1answer
55 views

Is there a reduction formula for $I_n=\displaystyle\int_{0}^{n\pi}\frac{\sin x}{1+x}\,dx$?

I haven't been able to manipulate this integral. I need to find the value of $I_n$ for $n=1,2,3,4$ and arrange them in ascending order.
0
votes
0answers
11 views

Correlation of an integral

Assuming that $a(x)$ is a random variable and correlation $\xi(x_1,x_2)=\langle a(x_1)a(x_2)\rangle$ is known (where angular bracket denote statistical averaging), it it possible to write the ...
1
vote
4answers
107 views

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$ $ $ This appears to be an easy problem, but it is consuming a lot of time, I am wondering if an easy way is possible. WHAT I DID : Wrote this as $\...
2
votes
2answers
32 views

Application Banach-Alaoglu Theorem

When reading about Banach-Alaoglu Theorem on Wikipedia, I read the following assertion: '' Let $f_n$ be a bounded sequence of functions in $L^p$. Then there exists a subsequence $f_{n_k}$ and an $f\...
1
vote
0answers
44 views

$\lim_{n \to \infty}\int_{0}^{a} e^{nx} dx$

Find $\lim_{n \to \infty}\int_{0}^{a} e^{nx} dx$ This seems to be a straightforward problem but since I am new to defnite integrals and thishas appeared in one the graduate exam papers I am looking ...
1
vote
0answers
43 views

Simplify an integral or the integrand involving hyperbolic functions

I would like to simplify the following integral or at least the integrand: $$f(t):=\int_{a}^{t-a} \frac{(\cosh(t-x)-\cosh(a))^{i\tau-1/2}}{(\cosh(x)-\cosh(a) )^{i\tau+1/2}} dx, \quad t>2a,$$ ...
1
vote
0answers
38 views

Why does Maple include x in the solution of this definite integral?

I have the following function defined in Maple: $$ f(x) := (2 - a + ax^2) \sqrt{1 + 4a^2x^2} $$ And I want to calculate the definite integral of this from -1 to 1: $$ \int_{-1}^{1}{f(x)dx} $$ I do ...
9
votes
2answers
109 views

Evaluating $\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$

How to calculate $$\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$$ I believe that we must use the Dirichlet integral $$\int_0^{\infty} {\frac{\sin{x}}{x}}\ ...
32
votes
1answer
794 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
0
votes
0answers
17 views

integrate equation

I am trying to integrate this equation, however I am not sure which method would be best. $\frac{\dot{a}}{a} = -2 \alpha \frac{\dot{M_1}}{M_1 + M_2}$ All the variables $a, M_1, M_2$ are time ...
2
votes
1answer
58 views

How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
0
votes
1answer
2k views

Integrating a solid using cartesian, cylindrical and spherical coordinates

The region $W$ is the cone shown below (see image). The angle at the vertex is $π/3$, and the top is flat and at a height of $7\sqrt{3}$. Write the limits of integration for $\int_W dV$ in the ...
1
vote
1answer
104 views
+100

On a solution to a triple integral.

I want to calculate the function $f(x,y,z) = z$ on the set $B = \{ (x,y,z) \in R^3 | z \ge \sqrt{7x^2 + 3y^2}, 2x + z \le 3 \}$ I tried to solve it without cylindrical substitutions. the solution is ...
0
votes
1answer
67 views

Find the limit $\lim_{x\to 0} x^{-3}\int_0^{x^2}\sin{(\sqrt t)}dt$

I use the fundemental theorem of calculus $$ \displaystyle\lim_{x\to 0}\frac{\displaystyle\int_0^{x^2}\sin{(\sqrt t)}dt}{x^3}=\frac{F_{(x^2)}-F_{(0)}}{x^3}="\frac{0}{0}" $$ Than apply L'hopital rule ...
4
votes
1answer
39 views

Conditions on a complex measure to be real

Let $(X,\mathcal{S}, \mu)$ be a measure space with $X$ a locally compact Hausdorff space, $\mathcal{S}$ the Borel subsets of $X$ and $\mu$ a complex measure. Suppose that $$ \int_X f \ d\mu \in \...
0
votes
0answers
41 views

How to calculate this integral $\int_0^a {dx\frac{{\tanh \left( x \right)}}{x}} $?

I encountered this integral, which Mathematica can't give an answer. $$\int_0^a {dx\frac{{\tanh \left( x \right)}}{x}} $$ I am sure the result contains Euler constant. How to do it?
2
votes
2answers
170 views

Nasty double integral with lots of exponentials

I am trying to compute a double integral. I will first define the functions that make up the integrand: $$F(\gamma)= A \,\exp(-a \, \gamma^ {1/2})+B \, \gamma^{-1/2}\left(1-\exp(-b\gamma^ {1/4})\...
9
votes
3answers
217 views

How to prove that$\int_{0}^{1}\ln{(x/(1-x))}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$

$$\int_{0}^{1}\ln{\big(\frac{x}{1-x}\big)}\ln{(1+x-x^2)}\frac{dx}{x}=-\frac{2}{5}\zeta{(3)}$$ Put $$\frac{x}{1-x}=y$$ $$I=\int_{0}^{\infty}\ln{y}\ln{(1+3y+y^2)}\frac{dy}{y(y+1)}=\frac{8}{5}\zeta{(3)...
2
votes
1answer
40 views

Calculate the area of a sphere drilled by two cylinders.

Let $S$ be the sphere given by the equation $x^2+y^2 +z^2 =4$ cut with $z \geq 0$. Now, we drill the semisphere that is left with two vertical cylinders of radius $1$, whose axes are respectively ...
0
votes
1answer
40 views

Integration in Maple 16

I need to solve this integral in Maple 16, I've tried various functions such as evalf, solve, ...
2
votes
2answers
47 views

Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
0
votes
0answers
17 views

Intuition/derivation for Cauchy's repeated integral formula?

https://en.m.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration I'm referring to this formula due to Cauchy. The wiki page has a proof, but what I'm looking for is a more direct derivation or ...
6
votes
0answers
98 views

Integral ${\large\int}_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx$

Could you please help me to find closed form expressions for the following definite integrals: $$I_1=\int_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx\approx0.3606065973884796896...$$ $$I_2=\int_0^{\pi/3}...
2
votes
0answers
28 views

How can I find the measure of $B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$?

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
2
votes
3answers
88 views

Integral $\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left({x} \right)\,dx$

Is there a closed form for this integral? $\displaystyle \int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left({x} \right)\,dx\\$ All I have been able to find, so far, is a numeric approximation of $-1.13348$
3
votes
1answer
130 views

Indefinite integral $\int x\sqrt{1+x}\mathrm{d}x$ using integration by parts

$$\int x\sqrt{1+x}\mathrm{d}x$$ $v'=\sqrt{1+x}$ $v=\frac{2}{3}(1+x)^{\frac{3}{2}}$ $u=x$ $u'=1$ $$\frac{2x}{3}(1+x)^{\frac{3}{2}}-\int\frac{2}{3}(1+x)^{\frac{3}{2}}=\frac{2x}{3}(1+x)^{\frac{3}{2}...
0
votes
1answer
18 views

Reference: Gaussianity of linear functional of Gaussian process

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets, If you take $X$ in $C([a,b])$ [i.e., $X$...
0
votes
0answers
30 views

Integrable and primitivable functions

Recently we learned the fundamental theorem of calculus and subsequently Leibniz-Newton formula, that linked the primitive with the definite integral. I know that Leibniz-Newton can be applied when ...
3
votes
2answers
743 views

What is $\int x\tan(x)dx$?

I have a problem when trying to solve this question Question. What is the answer of the indefinite integral $$\int x\tan x \; dx?$$ Maple gives a complicated answer based on the series. Is there any ...
2
votes
1answer
28 views

Calculating the Stokes Theorem

I was tasked with calculating $ \oint_{L}Fdr $ for when $F=xzi-j+yk$ (vetor form) and $$L = \begin{cases}z=5(x^2+y^2)-1 & \mbox{ } \mbox{} \\z=4 & \mbox{} \mbox{} \end{cases}$$ Using: ...
2
votes
1answer
39 views

What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)?

What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)? I took the numerator, used the change of coordinates $u = \alpha t$ and got that the limit was $\...
1
vote
1answer
35 views

What is $\int_{|\vec x| = 1, z \geq 0}(x^2+y^2)^pz^q$ for $p,q \geq 0$?

What is $\int_{|\vec x = (x,y,z)| = 1, z \geq 0}(x^2+y^2)^pz^q$ for $p,q \geq 0$? A hint is to use the Gamma function. I plugged in spherical coordinates and got: $I = 2\pi \int \int r^{2p+q+2}sin^{...
5
votes
3answers
79 views

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $?

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $ ? I tried substituting $x=1/t$ but that's making it more complicated.Any suggestions?
0
votes
0answers
14 views

Solving quasilinear PDE - 1D, time-dependant, convection

I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$ where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\...
1
vote
0answers
26 views

Differentiation under the integral sign and change of variables

Let $f \in C^2 \left(\mathbb{R}^2\right)$ with a bounded support, and let $f_\phi (x,y)=f(x\cos{\phi}-y\sin{\phi},x\cos{\phi}+y\sin{\phi}))$. Show that: $$\frac{d}{d\phi}\iint_{\mathbb{R}\times(0,\...
2
votes
1answer
38 views

Darboux integral epsilon-delta proof in piecewise continuous function

Using Darboux sums, if $f$ is a piecewise continuous function in $[a,b]$, then It is integrable in $[a,b]$. Given $\epsilon>0$, there is $\delta>0$ such that for every $P$ partition: $$||P||...
1
vote
0answers
26 views

Generalisation of $\int_0^1 \frac{1}{x^\alpha} dx< \infty \Leftrightarrow \alpha <1$ and a problem about the integral of a vectorial function.

1) In this problem I've only been able to prove the "trivial" implications, this is the one I need the most help with. $$ \int_{(0,1)^n} \frac{1}{x_1^{\alpha_1}+x_2^{\alpha_2}+...+x_n^{\alpha_n}} ...