Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
70,131
questions
3
votes
5
answers
141
views
Is there any other method to compute $\int_0^{\frac{\pi}{2}} \frac{x}{\sec x+\csc x} d x$?
Rationalization sometimes makes our life easier
Letting $x\mapsto \frac{\pi}{2}-x$ transforms the integral to
$\displaystyle I=\frac{\pi}{4} \int_0^{\frac{\pi}{2}} \frac{1}{\sec x+\csc x} d x=\frac{\...
- 11.7k
3
votes
1
answer
32
views
Integration a zero valued function product other function
Suppose I have a function $f(x) = 0$ for all $x$ except $0$. Now, $f(x) g(x)$ will also be zero for all $x$ except $0$ (or may be at $0$). Now, if try to find the value of $\int_{-\infty}^{\infty} f(x)...
- 363
0
votes
1
answer
45
views
Legendre polynomials: computing $\int_0^1 xP_n(x)\,dx$
I am trying to find the integration:
$$
\int_{0}^{1}{xP_n\left(x\right)}\,dx
$$
I know that I should split Rodrigues's formula up into two parts $P_{2n}\left(x\right)$ for even terms and $P_{2n+1}\...
- 1
6
votes
1
answer
164
views
$\lim_{n\to\infty} \left(\frac{e}{n}\right)^n \int_0^n |x(x-1)(x-2)\dots(x-n)|dx$
What is
$$
\lim_{n\to\infty} \left(\frac{e}{n}\right)^n
\int_0^n \left| x(x-1)(x-2) \cdots (x-n) \right| \, dx?
$$
Context: I was trying to find an asymptotic expression for the total area of the ...
- 8,418
0
votes
0
answers
47
views
change of variables formula in multiple integration
Use a change of variable in a double integral to compute the area of the square with vertices $(-3, 0)$, $(0, 3)$, $(3, 0)$, and $(0, -3)$. After the change of variable, the double integral should ...
- 11
1
vote
0
answers
17
views
Are there arguments for not using Cauchy Principal Value when there's odd singularity?
Question: Are there arguments or examples that shows I should not always use Cauchy Principal Value when Riemann's integral is not defined for mathematical applications?
I take the case which $f(x)$ ...
- 1,136
0
votes
0
answers
23
views
Du Bois-Reymond criterion for Riemann-integrability
Function f \in \textbf{R} ([a,b]) (1) \Leftrightarrow f is bounded on [a,b] and for all ...
0
votes
1
answer
78
views
Integral Identities for $S(x,y)=\sum_{k=1}^\infty\frac{y}{k^x(y+k)}$ and $\zeta(x)$ \ $\zeta(y)$
Define $$S(x,y)=\sum_{k=1}^\infty\frac{y}{k^x(y+k)}$$ This is a generalization of the harmonic function ($n=2$). There are many ways we could relate this to $\zeta(x)$. For example $$\lim_{y\...
- 3,394
1
vote
0
answers
73
views
What are the strategies to deal with intractable integrals encountered when solving ODE using variational method.?
I am trying to solve a nonlinear ODE: $$u_{xx}+\tan^2(x)u+gu^2=0$$ using the variational method, and I encountered an intractable integral. What are the strategies to deal with intractable integrals ...
2
votes
1
answer
37
views
Ratio of an increasing monotonic function and its integral is infinity
Inspired by the fact that for functions at the form: $f(x) = \frac{1}{x^\alpha}$ where $\alpha \ge 1$, the intgeral: $\int_{0}^{1} f(x)dx$ diverges to $\infty$ , and the ratio: $\frac{f'(x)}{f(x)}$ ...
- 186
2
votes
1
answer
31
views
Uniform convergence and improper integral
I was thinking about this claim (Real analysis question)
let ${f_n}$ be a series of continuous functions that converges uniformly to $f(x)=0$ on the interval $[1,\infty)$ and that satisfies the ...
- 31
-1
votes
1
answer
51
views
Calculate $ \int_{a}^{b} e^x \mathrm{d}x $with the aid of the Riemann subtotal.
Calculate $\int_{a}^{b}e^x \mathrm{d}x$ with the aid of the Riemann subtotal.
I know the Riemann subtotal and it is defined as follows: $S_n= \sum\limits_{i=1}^{n}{} f(ξ_i)\Delta x_i$. However, I have ...
- 1
3
votes
1
answer
60
views
Show $\int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi$
I am tring to prove
$$
\int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi.
$$
Numerical integration in Mathematica (truncating the integration bounds on $...
- 5,344
0
votes
0
answers
14
views
Meaning of boundary = 0 for a singular k chain: Two interpretations
For a singular k-chain - denoted by $s^k$ - there is a corresponding definition of a boundary, that is:
$$\partial s^k = \sum_j l_j \partial c_j^k,$$
wherein $l_j$ is an integer coefficient and the $...
- 313
4
votes
1
answer
57
views
Is there any algebraic manipulation to evaluate $\int \big( f(x)+af'(x) \big) e^x \text{d}x$, where $a$ is a real parameter $\ne 1$?
I knew that
$$\int \big( f(x)+f'(x) \big) e^x \text{d}x = e^x f(x) + c$$
For instance,
$$\int \big( \sin(2x)+2\cos(2x) \big) e^x \text{d}x = e^x \sin(2x) + c$$
Also,
$$\int \big( \sin(x)\cos(2x)+\cos(...
- 4,545
0
votes
4
answers
207
views
Prove that this expression is equal to $\pi$
Today, on the auspicious $\pi$ day, I saw on a local chat group $$\pi=4\phi^2\left(\phi^2+2\sqrt{\phi}\right)\left(\int_{\ 0}^{\ \infty\ }e^{x^2}\frac{\ \sin\left(x^2\right)\ }{x^2}dx\right)^2$$ I ...
1
vote
1
answer
58
views
Double integral where integration limit of the inner integral is integration variable of outer integral
I am trying to replicate a paper. I want to prove that:
$$\sigma^{-1} \int_t^T k \int_s^T e^{-\rho (z-s)} x(z) dz ds = k \int_t^T m(z-t) x(z) dz$$
where $m(s) = (\sigma \rho)^{-1} (1-e^{-\rho s})$. In ...
- 193
0
votes
0
answers
25
views
What's the meaning of the integral of the derivative of a distributin function $F$ if $F'(x)$ exists a.e.?
Lemma 2.2 on page 37 of Allan Gut's Probability: A Graduate Course (2nd edition) says :
Let F be a distribution function. Then:
(a) $F'(x)$ exists a.e., and is non-negative and finite.
(b) $\int_{a}^{...
- 11
7
votes
2
answers
130
views
Integral inequality implies majorization by solution of ODE
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing (and not necessarily differentiable nor continuous) and satisfy $$f(t)\leq f(0)+C\int_{0}^{t}f(s)^{2}ds,$$ where $C>0$. Is it true then that $$f(...
- 295
5
votes
4
answers
232
views
Can we evaluate the integral using Feynman’s, Frullani’s or double integral without Beta functions?
Background
When I first met the integral
$$I=\int_0^{\infty} \frac{\tan ^{-1}(a x)-\tan ^{-1}(x)}{x} d x,$$
I tried to use Feynman’s Technique by considering the integral with parameter $a$
$$ I(a)=\...
- 11.7k
5
votes
2
answers
135
views
Evaluate the integral $\int \frac{\cos^2(x)}{1-\beta \sin(x)}\mathrm{d}x$
Evaluate the integral $\int \frac{\cos^2(x)}{1-\beta \sin(x)}\mathrm{d}x$
Using Wolfram, I get a very complex result. Obviously the integration for
$$
\int \frac{\cos^2(x)}{1-\sin(x)} \mathrm{d}x= x-\...
- 53
3
votes
1
answer
84
views
Prove all rectificable functions on $[a,b]$ are integrable
Let $f:[a,b]→R$ be a rectificable function (means that $\mathrm{Length}(f,[a,b])$ is finite), $[a,b]\subset\Bbb R$, prove that $f$ is integrable.
$\mathrm{Length}(f,[a,b])=\sup\{\mathrm{Length}(f,X):...
- 117
6
votes
2
answers
90
views
Find the volume of the solid $Q$ cut from the sphere $x^2 + y^2 + z^2 ≤ 4$ by the cylinder $r = 2 \sin θ$ .
I set up the integral as $$\int_0^{\pi}\int_0^{2\sin\theta}\int_{-\sqrt{4-r^2}}^{\sqrt{4-r^2}} r dzdrd\theta$$ and got $\frac{16\pi}3$, but the answer is $\frac{16(3\pi-4)}9 \approx 9.64$, which is ...
2
votes
0
answers
31
views
Finding norm of integral operator on C[0,1] [duplicate]
I need help with the following:
Find norm of inegral operator $K:C[0,1] \to C[0,1]$ which is defined as
$$Kf(x) = \int_0^1 k(x,y)f(y)dy, \forall f \in C[0,1], x \in [0,1],$$
where $k \in C[0,1]^2$.
...
- 161
8
votes
5
answers
225
views
Integrate $\int \sqrt{(x+a)(x+b)} \space dx$
Integrate $\int \sqrt{(x+a)(x+b)} \space dx$
I've tried to use first Euler substitution:
$$\sqrt{x^2 + (a+b)x + ab} = x + t \implies x = \frac{t^2 - ab}{a + b - 2t}$$
$$dx = \frac{-2t^2 + 2(a+b)t - ...
- 593
0
votes
0
answers
36
views
Integral of $(T+dt)^{4}$
I want to take the anti derivative of $\frac{dq}{dt}=\beta \Delta T - (T+ \Delta T)^{4}$. I'm not sure how to go at it. I want to put it as $\frac{dq}{dt}=\beta dT - (T+ dT)^{4}$. I am worried about ...
1
vote
0
answers
82
views
Prove that $\int_0^1 f(x)g(x)\,dx \ge \int_0^1 f(x)\,dx \cdot \int_0^1 g(x)\,dx$
How to prove that if $f$ and $g$ are continuous and increase monotonically on $[0,1]$, then
$$
\int_0^1 f(x)g(x)\,dx \ge \int_0^1 f(x)\,dx \cdot \int_0^1 g(x)\,dx \quad ?
$$
The integral is understood ...
- 362
1
vote
1
answer
27
views
Integration of a bivariate domain to obtain the cumulative distribution of a bivariate function
Compute the (cumulative) probability function of $U = Y_1 - Y_2$, given that the joint density probability function $f(y_1, y_2)$ is
$$
\begin{align}
f(y_1, y_2) &= e^{-y_1} \qquad \text{in} \...
- 191
0
votes
2
answers
77
views
Issues with differentials [closed]
Can somebody explain what is the difference between
$\mathrm{d}^2x$
$(\mathrm{d}x)^2$
$\mathrm{d}x^2$?
For example, is $\displaystyle \dfrac{\mathrm{d}^2f}{\mathrm{d}x^2}$ the same as $\displaystyle ...
- 25
0
votes
0
answers
38
views
Relation between Lebesgue Integration and Integration on Differential Forms
In this paper Tao explains that "The unsigned definite integral generalises to the Lebesgue integral, or more generally to integration on a measure space. Finally, the signed definite integral ...
- 4,411
1
vote
1
answer
30
views
Integration over angular spherical coordinates
I am trying to solve the following integral in spherical 3D coordinates
$$\mathcal{I}(\theta,\phi)=
\int_0^{2\pi}d\phi_{\Delta}\int_{\delta}^{\Delta/2}\sin\theta_{\Delta}d\theta_{\Delta}
\frac{1}{[1-\...
- 189
0
votes
0
answers
34
views
3D Fourier transform on the Single Particle Green's Function
I'm having some troubles understanding how to use the 3D Fourier transform on this particular example:
$$(-\nu ^2 + \nabla ^2) G(\vec{x}) = \delta ^3(\vec{x})$$
Applying a 3D Fourier Transform:
$$(-\...
- 471
0
votes
1
answer
42
views
Prove $\int_1^\infty {f(x)-L\over f(x)} dx$ diverges
Let $f: [1, \infty)\rightarrow \mathbb R$ be a strictly decreasing function that converges to $0$ and fix some $L \in \mathbb R^+$. I want to show that $$\int_1^\infty {f(x)-L\over f(x)} dx$$ diverges....
- 2,006
2
votes
1
answer
85
views
How to Integrate the Gamma Function
Problem: Evaluate $$\int \Gamma(x)\, \mathrm{d}x$$
In asymptotic analysis, functions are compared with each other in terms of growth.
I want to know how much better one function is than another.
To ...
0
votes
2
answers
67
views
What is the natural length of the spring? - Hooke's Law
Question: A spring requires 5 Joules to stretch the spring from 8 cm to 12 cm, and an additional 4 Joules to stretch the spring from 12 cm to 14 cm. What is the natural length of the spring?
I have ...
- 25
-1
votes
0
answers
25
views
How to apply the Newton-Leibniz rule for the following when the integrand is a function of the upper limit? [closed]
I want to know the monotonicity w.r.t. $\epsilon_1$, i.e. $\frac{\partial}{\partial\epsilon_1}$of $\int_{0}^{\epsilon_1} \frac{F(\epsilon)}{\epsilon_1}d\epsilon$. In this problem, the integrand's ...
0
votes
1
answer
54
views
Solving an antiderivative via u-substitution
I have a density function defined as $f(x) = (1/10) \exp(-x/10)$. The answer to the solution suggests that the antiderivative, the cumulative distribution function, ought to be $\exp(-x/10)$, however ...
- 47
0
votes
0
answers
28
views
Counter intuitive result : a change of parameter in a product of functions not parameter invariant?
Say I have this function
$$
\int f(x) dx
$$
I now change $f(x)$ with $g(y) = f(x) \frac{dx}{dy}$
$$
\int g(y)\frac{dy}{dx}dx = \int g(y) dy
$$
So far, so good?
Say I have
$$
\int f(x) f(x) dx
$$
With ...
- 2,201
4
votes
0
answers
108
views
+50
Better understanding of integration of differential forms.
On page $100$ of Spivak's Calculus on Manifolds the following definition is made:
If $\omega$ is a $k$-form on $\mathbb{R}^k$, then $\omega = f\ dx_1\land\ldots\land dx_k$ for a unique function $f:\...
- 4,411
0
votes
1
answer
88
views
Why is $u=e^x$ not working for $\int 3e^{3x}\mathrm{d}x$?
$$\int{3e^{3x}}dx$$
Setting $u=e^x$ gives $$\frac{3}{e^x}\int{u^3}du,$$ which gives $\frac{3e^4}{4}+C$, which is evidently incorrect.
However, when I set $u=3x$, I get $$\int{e^u}du,$$ which ...
- 29
1
vote
1
answer
55
views
Improper Integral which should diverge
Consider
$$\int_{-2}^1 \frac{\text{d}x}{x} = \left(\lim_{\epsilon \to 0} \int_{-2}^{0-\epsilon} + \int_{0+\epsilon}^1\right) \frac{\text{d}x}{x} = \lim_{\epsilon \to 0} (\ln|-\epsilon| - \ln|-2| + \ln|...
- 1,975
3
votes
5
answers
98
views
Multiple cases within an integral after a $u$-substitution
The problem asks to solve this indefinite intregral:
$$I=\int{\frac{dx}{x\sqrt{x^2+1}}}$$
I did the following substitution (using $t$ as substitute value):
$$x^2+1=t\implies x=\pm\sqrt{t-1}$$
$$2xdx=...
- 143
0
votes
0
answers
44
views
Reference for proof of these 2 results in integration on Manifolds [closed]
These 2 questions were left as an exercise in my course on Differential geometry. I am an undergrad student and I am having extreme difficulty in following this particular course.
I was wondering if ...
- 69
0
votes
1
answer
33
views
A Riemann Sum clarification
I have to find the area under the curve $f(x) = x^2$ in the segment $[-2, 1]$ using Riemann Sum. So here is what I did, but it's wrong and I need some clarification about (see the questions in the end)...
- 1,975
4
votes
1
answer
119
views
Is this formula true?
There's a very simple formula in the book I am currently reading: $$\sum\limits_{a<x\leq b}f(x) = \int\limits_{a}^{b}f(x)\mathrm{d}x + \rho(b)f(b)-\rho(a)f(a) - \int\limits_{a}^{b}\rho(x)f'(x)\...
- 426
3
votes
1
answer
178
views
Evaluate $\int_{0}^{\frac{\pi}{4}} \sin\left(\arccos{\left(\frac{\cos{x + 5}}{1 + 2\cos{x}}\right)}\right) \,dx$
Evaluate the following integral $$\int_{0}^{\frac{\pi}{4}} \sin\left(\arccos{\left(\frac{\cos{x + 5}}{1 + 2\cos{x}}\right)}\right) \,dx$$
Working:
We get rid of the $\sin$ by considering the triangle. ...
- 688
1
vote
1
answer
54
views
Evaluate the following integral from $0$ to $\pi$
Evaluate $$\int_0^\pi\mathrm{e}^{\left|\cos\left(x\right)\right|}\sin\left(x\right)\left(2\sin\left(\dfrac{\cos\left(x\right)}{2}\right)+3\cos\left(\dfrac{\cos\left(x\right)}{2}\right)\right)\mathrm{d}...
2
votes
1
answer
103
views
Evaluating $\lim_{\lambda \rightarrow 0}\bigg(\int_0^1 (\beta x + \alpha(1-x))^\lambda dx\bigg)^{1/\lambda}$
Given $0 < \alpha < \beta$, I want to evaluate the limit of the integral
$$\lim_{\lambda \rightarrow 0}\bigg(\int_0^1 (\beta x + \alpha(1-x))^\lambda dx\bigg)^{1/\lambda}$$
I attempted u-...
- 2,006
2
votes
0
answers
21
views
Possibility of computing antiderivative using dual numbers
It is known that, given a function $f(x)$, plugging in the dual number $x+\varepsilon$, where $\varepsilon^2=0$, yields $f(x) + f'(x)\varepsilon$. For example: $$f(x) = x^3\\f(x+\varepsilon)=(x+\...
- 75
1
vote
2
answers
68
views
How to integrate the following definite integral involving trigonometric function?
How to integrate the following integral, which I have no idea. Thanks for your attention.
$$I=\int_0^{2\pi}d \theta \int_0^{\pi} e^{\sin\varphi(\cos\theta -\sin\theta)} \sin\varphi\, d \varphi$$
I ...
- 772