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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3answers
27 views

Calculus (what is y when x is?)

Given $y>0$ and $$dy/dx = (3x^2+4x)/y$$ If the point $(1,rad10)$ is on the graph relating x and y, then what is $y$ when $x=0$? I'm not sure whether or not to integrate, or just plug in the ...
-1
votes
4answers
35 views

Calculus (limits)

Compute $$\lim_{t\to0}\frac{\tan\left(\dfrac {1}4\pi + t\right) - \tan\left(\dfrac{1}4\pi\right)}t$$ Alright, so I'm taking the derivative first. Is there an easier way to take the derivative of ...
0
votes
1answer
50 views

Calculus (advanced integration)

Compute $\int (5^x+2e^{5 \ln x})dx$ The $5\ln x$ part confuses me. So far I have $5^x/\ln 5\:\:$
1
vote
0answers
78 views

Problem with trigonometric substitution proof

I'm sad, I can't get it. I know perfectly how to integrate using the mechanical process described in the books, but I want to understand the proof of it. My book (Stewart) says: In general we can ...
5
votes
3answers
147 views

Calculus (Integration)

Is there a simple way to integrate $\displaystyle\int\limits_{0}^{1/2}\dfrac{4}{1+4t^2}\,dt$ Original image at http://i.stack.imgur.com/DT6HF.gif I have no idea how to go about doing this. The ...
1
vote
2answers
31 views

Integration by parts

Integrate using integration by parts: $F(y) = (y+1)e^{-y}$ Find: Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts. Thus far, I've distributed the $e^y$ term and split ...
0
votes
1answer
29 views

Generalized Logarithmic Integral - reference request

This page at I&S forum defines the Generalized Logarithmic Integral as $$L\left[ \begin{matrix} a,b,c \\ d,e,f \end{matrix};z\right] =\int_0^z \frac{\log^a x \log^b(1-x)\log^c(1+x)}{x^d (1-x)^e ...
1
vote
1answer
52 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
1
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0answers
18 views

Counterexample for necessary condition of integrability

Can you give me an example of a non-negative function on $[0,1]$ that is NOT integrable, but $\lim_{t \to \infty} t \mu\{x : |f(x)| \geq t \} =0$?
2
votes
1answer
39 views

Double Integral of a piecewise function

If $F(x,y)$ is defined as $F(x,y) = x+y$ when $0 < x + y < 1$ and $0$ elsewhere, then find $$\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} F(x, y) \,dx \,dy$$. Math note: I've ...
1
vote
1answer
20 views

Expected length of a random vector

I meet a basic definition about the expected length of a random vector when reading a paper: What is "expected length" How to roughly derive both equations (yellow part) (Is that Gamma ...
2
votes
1answer
53 views

Indefinite integration of $1/\sqrt{3-5x-2x^2}$

Cannot make it out. $$\int \frac{dx}{(3-5x-2x^2)^{1/2}} $$ Is the problem correct, or does it have errors? I have a doubt.
2
votes
0answers
22 views

Normal Vector Affecting The Divergence Theorem

$\newcommand{\Div}{\operatorname{Div}}$I'm going to use an example to explain what I'm trying to ask. Let $T =\{(x,y,z): x^2+y^2=z^2, 0\leq z\leq3\}$, I'm asked to calculate $\iint_T ...
1
vote
2answers
53 views

Finding the integral of $(x^2+4x)/\sqrt{x^2+2x+2}$

Can somebody explain me how to calculate this integral? $$\int \frac{\left(x^2+4x\right)}{\sqrt{x^2+2x+2}}dx$$
1
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0answers
36 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration$$ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$$ I am not sure as to how to work with the branch cuts of both $\ln{x}$ and $\sqrt{1-x^2}$ ...
4
votes
1answer
66 views

Integral of $1/(1+x \tan(x))^2$

How would you solve the following integral? $$\int \frac{1}{(1+x\tan(x))^2} dx$$ Any help would be appreciated.
0
votes
2answers
55 views

A identity relating a infinite series and a definite integral [duplicate]

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!
0
votes
2answers
60 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
4
votes
2answers
57 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
13 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
62 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
61 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
83 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
36 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
72 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
58 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
14 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
21 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 ...
2
votes
0answers
55 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
26 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 $C: x=t^2,y=t+1,z=2t-1,0<=t=<1$
4
votes
2answers
95 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 = ...
3
votes
6answers
131 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
28 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
29 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
52 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
83 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 ...
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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 ...
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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
55 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
55 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
166 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
43 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
32 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
52 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
69 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
173 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 ...