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 ...

learn more… | top users | synonyms (3)

1
vote
3answers
42 views

Is it possible to $\int \sqrt{\cot x}$ by hand

$$\int \sqrt{\cot x}{dx}$$ $$\int \sqrt{\frac{\cos x}{\sin x}}{dx} $$ Using half angle formula $$\int \sqrt{\frac{1-\tan^2 \frac{x}{2}}{2\tan \frac{x}{2}}}{dx}$$ But I am not getting any lead from ...
2
votes
1answer
399 views

Integral Equation without solution?

working on a physical problem I arrived at the following equation $$ y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$ and after some struggling (not that easy to apply the basic Laplace ...
1
vote
0answers
14 views

When an inner product of two continuous functions and one of them are given, how can I find the other one?

$f(p)$ is the inner product of $h(t,p)$ and $x(t)$. That is, $\int_{T} h(t,p)x(t)dt = f(p)$ When $h(t,p)$ and $f(p)$ are given, how can I find $x(t)$?
0
votes
3answers
55 views

How to find a derivative of $f(x)=\int_0^{x^2}e^{xt^{-2}}dt$

Let $$f(x)=\int_0^{x^2}e^{xt^{-2}}dt$$ I want to find $$f'(x)$$ I tried to use taylor expansion: $$e^{xt^{-2}}=\sum_{n=0}^\infty \frac {x^nt^{-2n}} {n!}$$ Indefinite integral gives, $$\int e^{xt^{-2}}...
1
vote
1answer
28 views

Help needed in solving integration

I want to solve the following integration $$\int_0^{\infty}[1-(\frac{1}{1+x})^M]x^{-\frac{2}{\alpha}-1}dx$$ where $M$ is a positive integer and $\alpha \geq 2$ My attempt: In my attempt I use the ...
23
votes
6answers
1k views

Conjectured value of $\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$

I was curious whether this integral has a closed form expression : $$\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$$ The integrand has a singularity ...
-2
votes
1answer
41 views

Prove the following integral identity

Does any one have any idea on how to prove the following: $$\int_0^Sf(z)\,\mathrm{d}z+a\int_0^Sf(z)\left(\int_0^zf(z_1)\,\mathrm{d}z_1\right)\,\mathrm{d}z+a^2\int_0^Sf(z)\left[\int_0^zf(z_1)\left(\...
3
votes
2answers
47 views

What is the geometrical meaning of the integral of a vector valued function?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is an integrable function. then $\int_a^b f(x)dx$ can be considered as the area between the graph and the x-axis. But what if $f:\mathbb{R}^n\rightarrow \...
1
vote
0answers
21 views

If $f(x)\geq 0$ for all $x \in [a,b]$ and $\alpha \in BV([a,b])$ is increasing , then $\int_a^bf d\alpha \geq 0.$

This is a proof verification question. Here, $\, f$ is continuous and $\alpha$ is of bounded variation. My only tools are the sums, for a given partition $P = \{a=x_0 < \ldots < x_n = b \}$ of $...
3
votes
3answers
103 views

$\int f(x)\,dx - \int f(x)\,dx$

which is true $$\int f(x)\,dx - \int f(x)\,dx = 0$$ or $$\int f(x)\,dx - \int f(x)\,dx=c\text{ ?}$$ with $c$ some arbitary constant. My intuition says that 'something' subtracted by itself is ...
1
vote
1answer
69 views

How to evaluate $\int\frac{dx}{(2\sin x+\sec x)^4}$?

I tried a lot but I am not able to get a start. Can anyone give me the start of this question $$ \int\frac{dx}{(2\sin x+\sec x)^4} \ ? $$
0
votes
1answer
45 views

Solution to the convoluted integral equation

A have the following equation: $$f(a)=\int_0^ag(x)f(x)\,dx,$$ where $g(x)$ is a known function. Is there any solution to $f(a)$ just in terms of $g$?
13
votes
4answers
506 views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

It appears that $$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$ (so far I have about $1000$ decimal digits to confirm that). After changing variable $...
0
votes
3answers
2k views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
0
votes
1answer
37 views

How to solve the following result in integration?

I want to prove the following inequality: $$\int_{a}^{b}{|\cosh(\sqrt{iw})|^2dw} \geq \int_{a}^{b} \sinh^2\big(\sqrt{\frac{w}{2}}\big) dw$$ How to solve above integral? I am unable to keep $\iota (...
2
votes
1answer
106 views
+50

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=\...
0
votes
1answer
23 views

Choice of the limits for multivariable integral

Let $A \subseteq \mathbb{R}^2$ a limited set bordered through $x=0, x=1, y=-1+x, y=1-x^2$. Rotate A around the y-axis and define this set with $B$. Calculate the integral $$\int_B y\,\mathrm{d}x\...
0
votes
0answers
24 views

Is it possible construct a well-defined series through integration by parts?

I have a function $$G=\int_0^S\mathrm{d}x\,f(x)g(x)\mathrm{e}^{-\int_0^x\mathrm{d}z\,g(z)}~.$$ If $f(x)$ was unity, the above integral could have been easily written as $$G=1-\mathrm{e}^{-\int_0^S\...
4
votes
1answer
41 views

Volume in zero dimensional space

Suppose $A\subset \mathbb{R}^n$ is a compact, convex and centrally symmetric set such that $(x_1,\ldots,x_n)\in A$ if $$ |x_1|+\ldots+|x_r|+2\left(\sqrt{x_{r+1}^2 + x_{r+2}^2} + \ldots + \sqrt{x_{n-1}^...
1
vote
2answers
24 views

If an integrable function is orthogonal to all derivatives, then is f a constant?

Suppose that I have a function in $f \in L^1(\mathbb{R})$ such that $$\int_{\mathbb{R}}f(x)v'(x)\,dx = 0$$ for all test functions $v$ which are smooth with compact support. Can I show that $f(x)$ is ...
0
votes
0answers
33 views

How to calculate limits in triple integral?

I have the next excersise of triple integrals: $f(x,y,z) = \cos(x + y + z)$, Limited by planes $x=\pi, y=\pi, z=\pi $ I have to find/determine the value for this. Specific Questions: How can I ...
0
votes
1answer
44 views

Generalised integral

I want to carry the notion of Riemann integration to a more general setting. I have already given the following axioms on area function defined on an arbitrary Cartesian product: Let $X$ and $Y$ be ...
0
votes
0answers
20 views

Obatin $\int_{\gamma_1}F\cdot dl =\int_{\gamma_2}F\cdot dl$

Let $F = (F_1,F_2)$ be a $C^1$ vector field such that all its components are continuously differentiable in $\Omega$. Assume that $\frac{\partial F_1}{\partial y}=\frac{\partial F_2}{\partial x}$ Let ...
0
votes
2answers
45 views

When integrating by subsitution, once you find the function $u$ and $du$, what function should I integrate?

I understand that whenever we have a function to integrate, and we decide to use the method of variable change, what function should we integrate. For example, suppose this problem: $ \int_{a}^{b}\...
2
votes
1answer
33 views

Integration over the Haar measure of a compact Lie group preserves smoothness?

Let $G$ be a compact Lie group. Then there is a unique Haar (probability) measure on $G$. Let $f_g \colon G \to \mathbb{R}$ be a family of smooth functions $(f_g)_{g \in G}$, is the function $$ G \to \...
2
votes
1answer
41 views

Hadamard-like complex variable substitution

\begin{align} \frac\pi a &= \int_{-\infty}^\infty dxdye^{-a(x^2+y^2)}\\ \tag{1}&= \int_{-\infty}^\infty dxdye^{-a(x+iy)(x-iy)} \end{align} So far so good. Now introduce a complex variable $z$ ...
0
votes
1answer
52 views

How to solve this type of integrals($ \int_{|z| = 2} \frac{e^{2z}}{(z-2)^4} dz $)? [on hold]

How do you calculate such an integral? $$ \int_{|z| = 2} \frac{e^{2z}}{(z-2)^4} dz $$ Answer would be one of the following $$ A. \frac{8\pi i e^4}{3} $$ $$ B. \frac{\pi i e^4}{3} $$ $$ C. \frac{\pi ...
0
votes
2answers
46 views

Evaluate $\int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }}$

How do you solve this integral $$ \phi = \int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }} $$ ? Note: It appears in the Kepler problem and it should ...
3
votes
1answer
29 views

Convergence of a integral for every curve in the sphere

Let $S$ be the unit open sphere in $\mathbb{R}^3$: $x^2+y^2+z^2< 1$ and $\partial S$ its border $x^2+y^2+z^2= 1$. Let $f:S\cup \partial S\rightarrow \mathbb{R}$ be a continuous function which is ...
1
vote
1answer
50 views

Defined integral with min function.

I have to resolve an integral that I didn't see before and I can't find any examples online. So my integral is: $$\int_{0}^{\pi/2} \min(1, \tan (x)) dx $$ I have no idea what $\min(1, \tan x)$ means. ...
4
votes
3answers
343 views

Why is $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$?

I understand that $\delta(x)=0$ whenever $x \ne 0$ and that $\displaystyle\int_{x=-a}^{x=b} \delta(x) \, \mathrm{d}x = 1 \space$ $\forall a,b \gt 0$ and also $\displaystyle\int_{x=-\infty}^{x=\infty} ...
0
votes
3answers
87 views

Evaluate $\int \frac{\sqrt{64x^2-256}}{x}\,dx$

$$\int \frac{\sqrt{64x^2-256}}{x}\,dx$$ Image. I've tried this problem multiple times and cant seem to find where I made a mistake. If someone could please help explain where I went wrong I would ...
1
vote
1answer
34 views

Why do I get two answer when calculating this integral from two ways?

Assuming $a(t)=a_0\sin(\omega t)$, $v(0)=0$ and $x(0)=0$. I hope you know about basic relation between position, velocity and acceleration. They are derivatives of the proceeding one. I went on ...
0
votes
2answers
37 views

help me to find the triple integral

Use cylindrical coordinates to calculate for the given function and region: I found that the limits are for $x$ $0$ to $2\pi$ $r$ $0$ to $5$ and $z$ from $r^2$ to $25$ and the integration ...
3
votes
3answers
127 views

Help on how to show that $\int_{0}^{1}\left(2{x-1\over \ln^2{x}}-{x+1\over \ln{x}}\right)dx=3\ln{2}-2$

$$\int_{0}^{1}\left(2{x-1\over \ln^2{x}}-{x+1\over \ln{x}}\right)dx=3\ln{2}-2\tag1$$ Rewrite, so we can apply Frullani's formula on first part $$\int_{0}^{1}\left(-{x+1\over \ln{x}}+{2\over \ln{x}}+{...
0
votes
3answers
66 views

How to find a parametrization for a torus?

I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$ where $0<r<R$. I know I need to compute the metric tensor and ...
0
votes
2answers
37 views

Power Rule for Indefinite Integrals

To prove $\int x^p \, dx = \frac{x^{p+1}}{p+1} + C$, my calculus textbook writes: $$F '(x) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1} +C\right) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1}\right)+\frac{d}{...
0
votes
1answer
43 views

Derivation of an integration

Can someone explain to me the difference between the results of $ A$ and $B$, where $$A=\frac{d}{dc} \int_{-\infty}^c xf(x) dx $$ $$B= \frac{d}{dc} \int_c^{+\infty} xf(x) dx $$ You can image $f(x)$ ...
7
votes
3answers
538 views

Is there a nice way to find this integral $\int_0^1\frac{ \arcsin x}{x} \mathrm{d}x$?

$$\int_0^1\frac{ \arcsin x}{x}\,\mathrm dx$$ I was looking in my calculus text by chance when I saw this example , the solution is written also but it uses very tricky methods for me ! I wonder If ...
0
votes
1answer
24 views

Determine Integrability without use of Riemann Integral

Determine the function is integrable or not on its interval of definition: $f(x)=\begin{cases}0 \quad \textrm{if} \quad 0\le x\le1\\ x \quad \textrm{if} \quad 1\lt x \le2\end{cases}$ So in our class ...
7
votes
1answer
87 views

$\int \frac{dx}{\tan x + \cot x + \csc x + \sec x}$

$$\int \frac{dx}{\tan x + \cot x + \csc x + \sec x}$$ $$\tan x + \cot x + \csc x + \sec x=\frac{\sin x + 1}{\cos x} +\frac{\cos x + 1}{\sin x} $$ $$= \frac{\sin x +\cos x +1}{\sin x \cos x}$$ $$t= \...
12
votes
3answers
3k views

How to solve an definite integral of floor valute function?

How do you prove this identity: $$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$ I'd very much appreciate your help on this one!
13
votes
1answer
258 views
+50

A tricky integral - $\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt $

$$ \mathbf{\mbox{Evaluate:}}\qquad \int_{0}^{1} \sqrt{\frac{1}{\left(1 - t^{2}\right)^2} - \frac{\left(n + 1\right)^{2}\,t^{2n}}{\left(\, 1 - t^{2n+2}\,\,\right)^{2}}} \,\,\mathrm{d}t $$ where $n$ ...
1
vote
1answer
49 views

Iterated Integral with variable substitution

I need to calculate the double integral of the function $f(x,y) = (x+y)^9(x-y)^9$: $\int_0^{1/2} \int_x^{1-x} (x+y)^9(x-y)^9 dydx$ I have a solution but I definitely arrived at it after a sloppy ...
4
votes
1answer
144 views

What is relation between these integrals

I know $$ \int_{0}^{\frac{\pi}{2}}\ln(\sin x)dx=-\frac{\pi}{2}\ln(2)$$ What is relation between it and $$\int_{-\infty}^{\ln(4)}\frac{xe^x}{\sqrt{4e^x-e^{2x}}}dx$$ Please guid me. I have sixteen ...
2
votes
1answer
67 views

Volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$

Find the volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$, using integration. It is clear that this is not centered at the origin. So, how do I find the limits for an integral? Any suggestion ...
0
votes
3answers
128 views

Integral that makes square root of $\frac{\pi}{2}$ [duplicate]

My question is regarding a integral that´s giving me a huge headache. I want to show $$\int_{0}^{\infty}y^2e^{-\frac{y^2}{2}}dy=\sqrt{\frac{\pi}{2}}$$ I'm studying for an exam. I'm suppose to find ...
2
votes
3answers
108 views

Nice way to solve $\int\int \frac{1}{1-(xy)^2} dydx$?

This is something I've been thinking about lately; $$\int_0^1 \int_0^1 \frac{1}{1-(xy)^2} dydx$$ Solutions I've read involve making the substitutions: $x= \frac{sin(u)}{cos(v)}$ and $y= \frac{sin(v)}...
1
vote
2answers
60 views

How to evaluate this integral $\int_{-\infty}^{\infty}\exp(-ay^2)dy$ [duplicate]

I want to evaluate this integral $$\int_{-\infty}^{\infty}\exp(-ay^2)dy$$ using the error function definition. The problem I am facing is with the coefficient of $y^2$. Any suggestions? Fact $$...
3
votes
0answers
64 views

What approach should I use to solve integrals like this?

$$\int{\sqrt{1-{x^3}}}dx$$ I tried with $t=x^3$ but then I have the $3x^2$ dt that I can't get rid of.