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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0
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2answers
29 views

A identity relating a infinite series and a definite integral

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
-3
votes
0answers
23 views

Volume of $ \left|\frac{x}{a}\right|^n + \left|\frac{y}{b}\right|^n + \left|\frac{z}{c}\right|^n \leq 1 $

A solid is described by the equation $$ \left|\frac{x}{a}\right|^n + \left|\frac{y}{b}\right|^n + \left|\frac{z}{c}\right|^n \leq 1 $$ Find the volume of the solid Please give a proper proof ...
0
votes
1answer
41 views

Integrate $\int \sqrt{1+\cos(t/2)} dt$

I am looking for a neat and smart way to do this. I tried by substituting $u = 1+\cos(t/2)$ But I think its not the simplest way
1
vote
0answers
25 views

$\int \sqrt{1+\sin ^2 x} dx$ an elliptic integral?

It seems to be an elliptic integral of the second kind, but when $k=i$? This is going by the definition that $E(\theta,k)=\int_{0}^{\theta} \sqrt{1-k^2 \sin^2x}dx$. That seems a bit off. Or is this ...
1
vote
1answer
11 views

line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
3
votes
0answers
56 views

How prove this integral inequality$ \int_{0}^{+\infty}\frac{1}{x^x}{\rm d}x<2 $

Show that$$ \int_{0}^{+\infty}\frac{1}{x^x}{\rm d}x<2 $$
3
votes
1answer
48 views

How do I integrate $\int_{0}^{\frac{\pi^2}{4}}7\sin(\sqrt{x})dx$?

So, quick backstory. My semester just started and we are starting off by learning integration by parts. Which hasn't caused me much trouble except for this problem. ...
2
votes
2answers
71 views

Integral of ln(x)sech(x)

How can I prove that: $$\int_{0}^{\infty}\ln(x)\,\mbox{sech}(x)\,dx=\int_{0}^{\infty}\frac{2\ln(x)}{e^x+e^{-x}}\,dx\\=\pi\ln2+\frac{3}{2}\pi\ln(\pi)-2\pi\ln\!\Gamma(1/4)\approx-0.5208856126\!\dots$$ I ...
0
votes
0answers
32 views

estimation of an integration?

How to integrate the following expression, given $\int \rho(x)dx=M>0$, $\rho\geq 0$, and $\omega$ is a function? \begin{equation} \int \frac{x_1y_2-x_2y_1}{2\pi|x-y|^2}\rho(x)\omega(y)dxdy ...
2
votes
3answers
34 views

Approximation by definite integrals

I've seen a statement that says if $f$ is decreasing and continuous, then we have the following relation between the sum and integral: $$ \int_a^{b+1} f(x)dx \leq \sum_{i=a}^b f(i) \leq \int_{a-1}^b ...
2
votes
0answers
61 views

Integrate undefined function for $x$?

Can anyone help me to find to do the integral below? $$\int f(x+t)~f'(x-t)~ dx$$
2
votes
2answers
56 views

Indefinite integral of $\frac{\ln(x)}{(x-3)^2}$

I am trying to compute the integral $$ \int\frac{\ln(x)}{(x-3)^2}\mathrm{d}x $$ I have tried the following substitution, but seem to get nowhere: $u = x - 3$. $$x=u+3$$ $$dx=du$$ ...
0
votes
0answers
13 views

integral of a product of binomial series

I have to get a solution for this integral: $E_{n,k} = \int\limits_0^1 \! p \cdot \begin{pmatrix} n \\ k \end{pmatrix} \frac{(1-p^2)^k \cdot (3+p^2)^{n-k}}{4^n} \, dp$ Where... $k,n \in \mathbb ...
0
votes
0answers
11 views

Why does QUADPACK only enforce the least strict error boundary?

According to this reference (which is in agreement with my own numerical experiments), QUADPACK tries to fulfill the following accuracy requirement on the approximation error: |RESULT - I| $\le$ ...
2
votes
1answer
19 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
1
vote
0answers
51 views

How to evaluate the following two integral combined with anti-trigonometric function and trigonometric function?

\begin{align*} &\int_0^{\frac{\pi }{3}} {\arccos \frac{{1 - \cos x}}{{2\cos x}}dx} \\ &\int_0^{\frac{\pi }{2}} {\arccos \sqrt {\frac{{\cos x}}{{1 + 2\cos x}}} dx}. \end{align*} A few days ...
1
vote
1answer
32 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
-1
votes
1answer
25 views

Vector Calculus-integration [on hold]

$F(x,y,z)=(2xz+y^2)\hat{i}+2xy\hat{j}+(x^2+3z^2)\hat{k}$ Determine the work done by F to move the particle along the curve $C: x=t^2,y=t+1,z=2t-1,0<=t=<1$
3
votes
2answers
80 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
2
votes
6answers
109 views

Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but am unable to proceed further. I think the following integral can be ...
2
votes
0answers
27 views

McShane vs. Henstock-Kurzweil: Lebesgue integrable

Put in words, is it right to say that the difference of the McShane integral to the Henstock-Kurzweil integral is that the tags are not required to lie within $x_i\leq t_i\leq x_{i+1}$? If so, is ...
0
votes
1answer
28 views

Fourier transform of $F(x)=\exp(-x^2/(2 \sigma^2))$

I am looking for the fourier transform of $$F(x)=\exp\left(\frac{-x^2}{2a^2}\right)$$ where over $$-\infty<x<+\infty$$ I tried by definition $$f(u)={\int_{-\infty}^{+\infty} ...
0
votes
2answers
49 views

Calculate the mass of K

The curves $y = \cos x$ , $y = \sin 2 x$ and the y-axis defines the flat figure K. K's density in the point (x,y) is $\cos x$ mass units per area unit. Calculate the mass of K. I stated the density ...
3
votes
1answer
74 views

When may we ignore the limits of integration?

When we try to evaluate an integral such as, say $$\int_a^b{f(x)dx}$$ there is often the case that we can analytically find $$\int{f(x)dx}$$ a little faster (imagine leaving away the evaluation ...
0
votes
1answer
21 views

Antiderivative of unbounded function?

One way to visualize an antiderivative is that the area under the derivative is added to the initial value of the antiderivative to get the final value of the antiderivative over an interval. The ...
-1
votes
0answers
36 views

Darboux Integrable Functions vs. Uniform Closure of Simple Functions

Is there a Darboux integrable function $f:[0,1]\to E$ with values in a Banach space $E$ that is not the uniform limit of simple functions $s=\sum_\alpha\chi_{A_\alpha}$ taken over finite sums with ...
0
votes
0answers
54 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
2
votes
2answers
53 views

Show that $H_i=H_{n-i}$ and $\sum H_i=1$

We define $$H_i=\frac{1}{n}\frac{(-1)^{n-1}}{i!(n-1)!}\int_{0}^{n}\prod_{j=0,j\neq i}^{n}(x-j)dx$$ This is called the Newton-Cotes coefficient. Here is the exercise: First, convince yourself that ...
8
votes
2answers
155 views

Integral inequality: $\def\intd{\,\mathrm d}\int_a^b(f'(x))^2\intd x-2\big(f(a)+f(b)\big)^2\geq\frac8{(b-a)^2}\int_a^b(f(x))^2\intd x$

I have a problem which I think is wrong. Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with $f'$ continuous such that $$\int_a^b f(x) \intd x = f\left(\frac{a+b}{2}\right) = 0$$ ...
0
votes
1answer
42 views

Integral of a bivariate normal cdf

Let $$ \Phi_2(x,y;\rho):=\int_{-\infty}^y\int_{-\infty}^x \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(s^2+t^2-2st\rho)} \, ds \, dt $$ be the joint cdf of bi-variate normal random ...
0
votes
0answers
31 views

Find a Harmonic conjugate $v(x,y)$ to $u(x,y)$.

Show that $u(x,y) = \frac{y^2}{x^3+y^3}$ in some domain and find the harmonic conjugate $v(x,y)$ to $u(x,y)$.
2
votes
2answers
42 views

Find the work done by the force field in moving the particle from one point to another

Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$ This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$ ...
4
votes
1answer
50 views

Can you prove a definite integral has no closed form?

It is a well known fact that some functions posses no closed form antiderivative yet still they have definite integrals that have a closed form. A classic example is the Gaussian integral ...
0
votes
2answers
68 views

A limit of an Integral

Consider the following limit $$K=\lim_{x\rightarrow \infty}\frac{1}{x(1-x)}\left(1-\int_{\mathbb{R}}g(y;x)^x f(y)^{1-x}\mathrm{d}y\right)$$ where $f$ and $g$ are any continuous probability density ...
10
votes
3answers
168 views

How do I integrate $\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}$

How do I evaluate this indefinite integral, for $|k| < 1$: $$ \int\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}\mathrm{d}x $$ I tried the change of variable $t=\sin x$, and obtained two integrals, but I ...
4
votes
1answer
265 views

How do I proceed with this integral?

I have the following integral: $$\int \frac{\tan^{-1}(\ln (x))}{x}dx.$$ Trying to solve it by integration by parts (with $u=\ln (x)$ and $v=\tan^{-1} (\ln (x))$, I have seemingly come to a dead end: ...
4
votes
3answers
216 views

The Absolute Value in the Integral of $1/x$

$$\int\frac{1}{x}dx=\ln| x |+C$$ Why the absolute value? Why is the following not valid: $$\int\frac{1}{x}dx=\ln x+C$$
0
votes
1answer
28 views

Taking the integral of one over x

Given: $$ \int\frac{1}{x}dx = \ln |\ x\ | +C $$ We have: $$ 7\int\frac{1}{x}dx = 7\ln(x) + C $$ $$ \int\frac{1}{2x+4} = \frac{1}{2}\ln(2x+4) + C $$ $$ \int\frac{t}{t^{2}+4}dx = ...
0
votes
0answers
25 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
1
vote
1answer
17 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
0
votes
0answers
20 views

Is it a valid kernel?

Suppose we have a valid psd kernel $k(x, y)$, then we set $\psi(x, y) = ak(x, y) + bk(x, x_0) + ck(x_0, y) + dk(x_0, x_0)$ where $a, b, c, d \geq 0$. Thus, is $\psi(x, y)$ a valid psd kernel? As we ...
1
vote
1answer
34 views

Simpson's rule and Trapezoid Rule?

Let $S(n)$ and $T(n)$ be the approximations of a function using $n$ intervals by using Simpson's rule and the Trapezoid rule respectfully. My book then states: $$S(2n) = \frac{4T(2n) - T(n)}{3}$$ ...
0
votes
1answer
36 views

Partial fraction decomposition and polynomials?

This answer gives a really great explanation of why partial fraction decomposition works. However, the explanation implies that rational functions can be decomposed into a sum of fractions plus a ...
18
votes
3answers
292 views
+200

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: ...
1
vote
0answers
24 views

Is there a general technique to obtain asymptotic expansion of integral of the form $\int_0^{x} d x' f(x') $ as $x \to \infty$.

In particular, I wish to find a proof of $\int_{0}^{\infty} \dfrac{\sin(x)}{x} = \dfrac{\pi}{2}$ using such a method. Thanks in advance.
1
vote
2answers
28 views

Primitive for $f(x)=\frac{2+3x+x^2}{x(x^2+1)}.$

I have to find a primitive of $$f(x)=\frac{2+3x+x^2}{x(x^2+1)}.$$ I tried to use partial decomposition but I am having trouble to evaluate this fraction at $0$. Using this method we have ...
-2
votes
2answers
59 views

4 Integrals I Need Help With [on hold]

Working on these 4 problems for a review worksheet. I got up to this point and got stumped at #7. Can anyone explain which method to use for each problem? Once I know how to approach the problem I am ...
3
votes
2answers
64 views

cosine integral

Show that $$\int_0^x \frac{1-\cos(t)}{t}=\gamma+\ln(x)-\operatorname{Ci}(x)$$ where $$\operatorname{Ci}(x)=-\int_x^\infty \frac{\cos(t)}{t} \, dt$$ and gamma is an euler-mascheroni constant. I did as ...
0
votes
1answer
18 views

Derivative of an integral with variable in upper bound and a term of the integrand

So I want to take the first and second derivatives of a function g(Z) which is made up of several terms, one of which is where Z and H are our variables. Taking the derivative of this, it seems ...
1
vote
1answer
72 views

Prove that $\int_{-1}^1P_n^2(x)dx=\frac{2}{2n+1}$, where $P_n(x)$ is a Legendre polynomial.

Using Rodrigues' formula and integrating by parts $n$ times, prove that $$\int_{-1}^1P_n^2(x)dx=\frac{2}{2n+1}$$ where $P_n(x)$ is a Legendre polynomial. I tried this way Let $$f(x)=(x^2-1)^s$$ ...