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 ...

learn more… | top users | synonyms (2)

3
votes
2answers
139 views

Evaluate $\underbrace{\idotsint}_n \exp\left(-\sum_{1\le i\le j\le n}^n x_i x_j\right) \mathrm{d}x_1\cdots\mathrm{d}x_n $

I am currently studying V.I. Arnold's course, and I am stuck on this exercise: Evaluate $$ \underbrace{\idotsint}_{n} \exp\left(-\sum_{1\le i\le j\le n}^n x_i x_j\right) ...
1
vote
1answer
106 views

If $ I = \int_0^\pi \sin(\sin^{-1}\{x\})dx $ then what is $\big[I\big]$?

I was working on the following question: If $$ I = \int_0^\pi \sin(\sin^{-1}\{x\})dx $$ then what is $\big[I\big]$? ($ \{x\} $ means the fractional part of $x$) My solution went like this: ...
2
votes
0answers
60 views

If someone asked, and if I do t understand its soution, then, how do i understand? Do I not have aright to ask again? [duplicate]

First of all, I searched the question, and someone asked, I found its solution. But I think, that solution is not clear enough. Forvexample, there, the reason why the integral is zero is not ...
2
votes
2answers
134 views

How to solve this “simple” ordinary differential equation?

I am trying to learn more about calculus by myself, in order to be able to use dynamical systems analysis methods. In a book example, I have to find $f(t)$ from this: ...
6
votes
2answers
187 views

Vector Calculus Proof: $\oint_K \nabla f\cdot \vec n\,ds$ has two possible values

I'm looking over the last chapter in my University Calculus (2nd edition) text by Hass, Weir, and Thomas. I came across the following problem (not homework), with which I have had some difficulty. ...
2
votes
1answer
105 views

Integral of function with a pretty long name

I am looking for a compact result of this integral: $$\int_R^\infty k_n(x) \, dx,$$ where $k_n$ is the modified spherical Bessel function of the second kind (explanation to this function). ...
6
votes
4answers
449 views

For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$

Suppose $f:[0,1]\to \mathbb{R}$ is continuous. Show that  $$\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$$ My answer so far: First I want to assume that $f\in C^1$. Then  $$n\int_0^1f(x)x^n\,dx ...
11
votes
2answers
362 views

closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$

closed form of $$\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$$ for $a,b>0$ n=1 we get $$\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^1}=\frac{2\pi}{ab}$$ n=2 we get ...
3
votes
1answer
112 views

Calculate the integral $\int_0^{x} (\lfloor t+1 \rfloor)^3dt$

Here's the integral, $$\int_0^{x} (\lfloor t+1 \rfloor)^3dt$$ I have some knowledge of computing the integrals of discontinous functions but the cube function and the independent variable limit ...
2
votes
1answer
166 views

Definition clarification on orientation on a manifold.

I have been trying to self-learn differential geometry. I think I may have misunderstood/missed out on something along the way. It is said that for $X$ an $n$-form, $M$ a differentiable manifold, ...
36
votes
1answer
924 views

To evaluate $\int_0^{+\infty} \frac{\;dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+(\frac{a+b}{2})^2}\sqrt{x^2+ab}}$$ ...
4
votes
2answers
162 views

How to evaluate the integral $\int_{0}^{\infty}\frac{\sin{(ax)}}{x^b} dx$

Evaluate the integral $$\int_0^\infty \dfrac{\sin(ax)}{x^b} \, dx,\quad a\in \mathbb{R},\quad 0<b<2.$$ I know $a=1$ ,and $ b\in \mathbb{N}$, I can find the value, How to evaluate this integral ...
2
votes
4answers
441 views

Volume of cube section above intersection with plane

Suppose we have a unit cube (side=1) and a plane with equation $x+y+z=\alpha$. I'd like to compute the volume of the region that results once the plane sections the cube (above the plane). There are ...
2
votes
1answer
647 views

Double integral in polar coordinates

Use polar coordinates to find the volume of the given solid inside the sphere $$x^2+y^2+z^2=16$$ and outside the cylinder $$x^2+y^2=4$$ When I try to solve the problem, I keep getting the wrong ...
4
votes
3answers
262 views

show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $

show that $$\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2\left(\frac{a\pi}{2}\right) $$ also I think we can solve it by contour integration but how and its better ...
4
votes
2answers
108 views

show that $\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$

show that $$\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$$ I think we can solve it by contour integration but I dont know how. If someone can solve it by two way using complex ...
2
votes
2answers
114 views

Finding a definite integral by residue integration?

I have a probability distribution of the form $$\frac{k_1}{(k_2 x^2+k_3 x+k_4)^n}$$ and I want to find the mean and variance—but am running into problems with that. The mean would be ...
2
votes
1answer
148 views

Prove that $\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$

Let $f$ be a continuously differentiable function on $[0,1]$ and $f(0)=0$. Prove that $$\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$$ Thank you!
1
vote
0answers
50 views

Calculate $\int_\gamma(y^2+z)\,dx + z\, dy + xy\,dz$ where $\gamma$ is the intersection between two surfaces.

The problem I have is to find the normal vextor $\textbf{N}$, for Stokes's theorem, and then determine the area of integration. Here's the question: Let $\gamma$ be the intersection between the ...
1
vote
1answer
60 views

Numerical integration over an actual series, is this possible?

I was wondering about the following: Is there any clever way to evaluate integrals over a series? Let me choose an example: $$\int_{\frac{2\pi}{3}}^{\pi} \sum_{n=0}^{\infty} P_n(\cos(x))dx$$ where ...
6
votes
2answers
188 views

$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$ without using trigonometry?

$$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$$ Is there any way to find the answer without using trigonometry, like this? Hint by Parth Thakkat: $$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$$ $$ = \int ...
2
votes
0answers
71 views

Evaluation of oscillating gaussian integral

I've problems to evaluate the following integral $$\int_{\mathbb{R}^3} dx \, dy \, dz \, \frac{e^{-i \Gamma |\vec{r}-\vec{r_0}|}}{|\vec{r}-\vec{r_0}|} \frac{e^{i \Upsilon |\vec{r}|}}{|\vec{r}|} ...
3
votes
2answers
95 views

Find the volume of $K=\{(x,y,z):|x-z^2|+|y-z^2|+z^2\le1\}$

How can I find the volume of $K=\{(x,y,z):|x-z^2|+|y-z^2|+z^2\le1\}$? First, $z^2\le1$ since $|x-z^2|+|y-z^2|\ge0$ This would give me $$\int \limits_{-1}^1 \left(\iint\limits_K \,dx\,dy\right)\,dz$$ ...
1
vote
1answer
43 views

Where's the mistake in integrating $\operatorname{sech}$ using series?

$$\begin{equation} \begin{split} \int\frac{2dx}{e^x+e^{-x}} & = \int\frac{2e^{-x}dx}{1+e^{-2x}} \\ &= \int 2e^{-x} \sum_{k=0}^{\infty}(-1)^ke^{-2kx}\,dx \\ &= ...
2
votes
1answer
89 views

Evaluating $\int \frac{L}{E-iR}\operatorname d i$

Can someone please explain to me step by step, how to evaluate this integral? $E$, $R$ and $L$ are constants. $$\int \frac{L}{E-iR}di$$ The result should be: $-\frac{L}{R}\ln(E-iR) + C$ Thank you. ...
12
votes
3answers
368 views

integral of $\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$

what is the answer of $$\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$$ I saw the answer of $$\int \limits_{0}^{\infty}\left(\frac {\sin x} {x}\right)^ndx$$ but for my question i didn't see ...
7
votes
4answers
174 views

show that $\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$

show that: $$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$$ where $n=0,1,2,3,\ldots$. is there any help? thanks for all
1
vote
1answer
101 views

Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between ...
1
vote
0answers
33 views

Show that $\sum\limits_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$. [duplicate]

Prove that for $x,y$ positive integers, $$\sum_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$$ One way is to use the beta-gamma functions relation: ...
1
vote
2answers
53 views

how to calculate partial derivative?

how do I find $\frac{\partial q}{\partial k}$ of $q(k,l,m) = k\,p(k,l) + m^2$ ? I have tried $\frac{\partial q}{\partial k}= p(k,l) \times\begin{bmatrix}\frac{dk}{dm}+\frac{dl}{dm}\end{bmatrix} + ...
3
votes
3answers
140 views

Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$.

Let $I$ be a generalized rectangle in $\Bbb R^n$ Suppose that the function $f\colon I\to \Bbb R$ is continuous. Assume that $f(x)\ge 0$, $\forall x \in I$ Prove that $\int_{I}f=0 \iff$ the function ...
2
votes
4answers
224 views

How to integrate : $\sqrt{\frac{a-x}{x-b}}$

Problem : How to integrate : $\sqrt{\frac{a-x}{x-b}}$ Unable to find the substitution for this : $\sqrt{\frac{a-x}{x-b}}$ Please help how to proceed ...........thanks..
0
votes
0answers
33 views

estimates for sums

How can I prove the estimate $ \int\limits_0^{2\pi } \vert \sum\limits_{j=0}^k \ r^je^{ij \theta} \vert \,d\theta \leq \int\limits_0^{2\pi } {\vert \sum\limits_{j=0}^k \ e^{ij \theta} \vert ...
1
vote
4answers
321 views

Show that $f$ is integrable and $\int_{I} f=0$

Let $I$ b a generalized rectangle in $\Bbb R^n$ Suppose the bounded function $f:I\to \Bbb R$ assumes the value $0$ except at a single point $x \in I$ Show that $f$ is integrable and $\int_{I} f=0$ ...
5
votes
1answer
130 views

Differentiation under the integral sign??

When I tried to show this, I didn't get the integral of the derivative - only the other terms, and I have no idea why. Here's the working I have; $\dfrac{d}{dx} \displaystyle\int_{a(x)}^{b(x)} ...
3
votes
1answer
43 views

Evulating $\int_I f$ by using Darboux Sum convergence Criterion

I tried to solve the question. But, there may be some mistakes. I want to learn this properly. If there exist any notation mistake, typo, a general mistake in solution way or else, please correct ...
2
votes
2answers
127 views

Finding surface area - integral of $\sqrt{1+z^2}$

Sorry about this, this is more of a "am I going the right way" question, there's a surface it goes: $$x^2+y^2-z^2=1$$ Now this is nice because $x^2+y^2=r^2=1+z^2$ thus $r=\sqrt{1+z^2}$ (I want the ...
3
votes
2answers
99 views

Use the Integrability Criterion to show that the function $f: I \to \Bbb R$ is integrable.

Question: For the generalized rectangle $I= [0,1]\times [0,1]$ in the plane $\Bbb R^2$ $$f(x,y)=\begin{cases} 5 & if\ \ (x,y)\ is\ in\ I\ and\ x> 1/2 \\ 1 & if\ (x,y)\ is\ in\ I\ \ and\ ...
4
votes
2answers
116 views

Integration involving fixed points

A few days ago I ran into this statement If $a$, $b$ are fixed points of a function $f$, then $$\int_a^b(f(x) + f^{-1}(x)) \,\mathrm dx = b^2 - a^2.$$ I checked it for a few simple cases like ...
2
votes
0answers
64 views

Why write $\mathrm dx, \mathrm dt$ etc. at the beginning of an integral? [duplicate]

I've noticed that many people here (on Math.SE) as well as elsewhere write integrals out like this: $$\int^a_b \mathrm dt \; f(t)$$ instead of the more common (at least from what I've seen): ...
2
votes
1answer
69 views

show that $\int_{-\infty}^{\infty} \frac {(\sin x) (x^2+a^2)}{x(x^2+b^2)}dx=\frac{\pi(a^2+e^{-b}(b^2-a^2))}{b^2}$

show that $$\int_{-\infty}^{\infty} \frac {(\sin x) (x^2+a^2)}{x(x^2+b^2)}dx=\frac{\pi(a^2+e^{-b}(b^2-a^2))}{b^2}$$ for every $a,b>0$ thanks for all
5
votes
5answers
482 views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
0
votes
2answers
195 views

Evaluate $\int_{0}^{\infty} \frac{x^{1/2}}{1 + x^2}\,\mathrm dx$ using Residue Theorem.

Evaluate $\int_{0}^{\infty} \frac{x^{1/2}}{1 + x^2}\,\mathrm dx$ using the Residue Theorem. I have been given a formula to compute integrals of this type: $I = \int_{0}^{\infty} R(x) ...
4
votes
3answers
96 views

Show that if f is integrable on [a,b], then |f| is also integrable.

The problem suggests doing it by showing that $U(P,|f|) - L(P,|f|) \le U(P,f)-L(P,f)$ for some partition $P$. I can get the other steps after that, but I've tried proving this inequality on my own ...
2
votes
4answers
86 views

Testing for convergence of this function

For the integral $$\int_2^\infty \dfrac{x+1}{(x-1)(x^2+x+1)}dx .$$ Can I know if it's convergent or not? If it does can I know how to evaluate it? I tried to use $u$ substitution but it didn't ...
7
votes
1answer
147 views

Does this inner product on $L^1([0,1])$ have a name?

Math people: For $f, g \in L^1([0,1])$, define $$\langle f,g \rangle = \int_0^1 \int_0^1 f(t)g(t')\exp(-|t-t'|)dt'\,dt.$$ Although we don't normally think of $L^1([0,1])$ as an inner product space, ...
2
votes
2answers
58 views

Is there a 0-form $\tau$ with $d\tau=\omega$?

Consider the 1-form $$ \omega=(x^2-yz)dx+(y^2-xz)dy-xydz. $$ Does a 0-Form $\tau$ on $\mathbb{R}^3$ exist which fullfils $d\tau=\omega$? Hello, my simple answer is: YES, because ...
1
vote
1answer
146 views

Function zero almost everywhere

Assume $f$ is an integrable function on $\mathbb{R^n}$. Assume for every bounded continuous function g on $\mathbb{R^n}$, $\int_\mathbb{R^n}fg=0$. Prove $f$ must equal $0$ almost everywhere. I am ...
2
votes
1answer
58 views

Change of variables in 3 dimensions

Consider the following integral: $$\int_{|x| = \epsilon} \phi(x) \frac{e^{-m|x|}}{4 \pi |x|^2} d^3x.$$ I wanna show that this integral goes to $\phi(0)$ for $\epsilon \rightarrow 0$. The idea is ...
3
votes
1answer
204 views

Tough Legendre Integral

I am currently fighting with the following integral. I have simplified it to this one here: $\int_{-1}^{\cos(\alpha)} P_l(t)P_{l'}(t) dt$, where $P_l$ is the l-th Legendre polynomial. unfortunately ...