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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7 views

Evaluate the given integral along the given (positively oriented) circle.

Ok, so I have the following problems that I am working on. It says to evaluate 1) where C is given by |z+1|=1/2 2) where C is given by |z-2|=1/2 3) where C is given by |z|=2 4) where C ...
-3
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0answers
28 views

Find $ \int_{\theta_0}^{\theta} \cos \theta \left( \sin 2\theta \right)^{3/2} \, \mathrm{d}\theta $

Find $$ \displaystyle\int_{\theta_0}^{\theta} \cos \phi \left( \sin 2\phi \right)^{3/2} \, \mathrm{d}\phi $$
1
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4answers
48 views

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
-2
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1answer
34 views

Evaluating a complex integral using the Cauchy integral formula [on hold]

I need to evaluate the following integral counterclockwise: $$\oint_{\left | z \right |=\frac{1}{2}} \frac{dz}{(z-1)\sin z} $$ using the Cauchy integral formula
3
votes
2answers
23 views

Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the laplace transform of the function $f(t)=te^{-t}\sin(2t)$ using only the properties of laplace transform, meaning, use clever tricks and the table shown at ...
2
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0answers
9 views

Bounding $\int_{\infty}^{\infty}|g(s)v^3k(v)|dv$ where $k$ is a second-order kernel

Suppose $k$ is a nonnegative, bounded real-valued function that satisfies $$ \int_{-\infty}^\infty k(v)dv=1,\quad k(v)=k(-v),\quad \int_{-\infty}^\infty ...
1
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0answers
33 views

Joint and marginal distributions and expectations (Is my proof right?)

1. Please look this following proof first: 2. I want to proof the conditional case, and the proof process is following. 3. I want somebody to help to check whether my proof is right? Thanks
5
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2answers
57 views

Quadratic Expressions: Advanced techniques of Integration

$$\int \frac{x}{\sqrt{5+12x-9x^2}}\,dx$$ After two steps I arrive at $\displaystyle{ \int \frac{x}{\sqrt{9-(3x-2)^2}}}\,dx$ Using trigonometric substitution, we have a triangle with a cosine of ...
1
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2answers
18 views

Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
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0answers
30 views

Advanced Integration techniques: Quadratic Expressions and U-Substitution

Find $\int$ $ dx(2x-1)\over (x^2-6x+13) $ In the final steps after a u-substitution, one arrives at $\int$ $dx(2u)\over(u^2+4)$ + $\int$ $(5du)\over (u^2+4)$ The next step is arriving at ln(u^2+4) + ...
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1answer
23 views

Finding pathline

I've been trying to find the pathline of a particle dropped in a steady flow defined by the following vector components: $$ u= \frac{-2x}{(x^2+y^2+1)^2} \hat i + \frac{-2y}{(x^2+y^2+1)^2}\hat j $$ in ...
1
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2answers
46 views

Solutions to the integral $\int \frac {dx}{2\sqrt x (x+1)}$

I am given a question to solve the integral $\int \frac {dx}{2\sqrt x (x+1)}$. When I substitute $x+1 = t^2$, I get the solution as $\space \ln(\sqrt{x+1} + \sqrt x) +C$; while when I substitute ...
-3
votes
1answer
69 views

I do not understand the last step of this proof. [on hold]

1. PLEASE LOOK THE FOLLOWING PROOF FIRST. 2. Suzu explained the fist several steps to me in this page :Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$ . But I still ...
0
votes
2answers
27 views

Evaluation of an integral of some expressions involving fractions

I am stuck in evaluating the following integral: \begin{equation} \int_{0}^{b-a} \frac{1}{\sqrt{u} (a+u)} \,du, \end{equation} where $0<a<b$. Any ideas?
1
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1answer
25 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
1
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0answers
11 views

A treatise on Probabilistic arguments (or even Laplace/Fourier transforms) to solve limits/integrals from basic calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those ...
5
votes
4answers
159 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...
2
votes
1answer
34 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ (The limits of the integrals are all ...
4
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0answers
59 views

An infinite series of integrals $\int_{0}^{\eta}\cos nt\log\left(\frac{\cos(t/2)+\sqrt{\cos^2(t/2) -\cos^2(\eta/2)}}{\cos(\eta/2)}\right) dt$

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the series is an integral of a complicated transcendental function like the one in the title. There ...
3
votes
3answers
60 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
5
votes
4answers
118 views

Evaluating $\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$

Evaluating $$\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$$ using $ux=\sqrt{x^2-1}$ I try to $u^2x^2=x^2-1$ $x^2=\frac{-1}{u^2-1}$ However I cant get rid of $x$ because derivative has $x\;dx$. How can I ...
0
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1answer
42 views

How to compute $\int_0^1\int_0^1 |x-y|dxdy$? [on hold]

Can anyone help me to solve $\int_0^1\int_0^1 |x-y|dxdy$ ? Thanks.
5
votes
2answers
89 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
1
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0answers
88 views

Evaluation of $\displaystyle\int\frac{1}{x^4+1}$ (Spivak's Calculus, Chapter 19, Problem 6viii) [duplicate]

Solve the integral:$$\int \frac{1}{x^4+1}\mathrm{d}x$$ I tried the substitution $x=\tan\theta \Rightarrow dx=\sec^2 \theta\,\mathrm{d}\theta$, but that leads to a dead end. $$\begin{align}\int ...
0
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0answers
35 views

Changing integral limits [on hold]

I am going through some algebra for my dissertation and in the algebra the limits of integration are changed. For example they went from x to infinity but they changed them to y going to infinity. ...
7
votes
2answers
98 views

How find this integral $I=\int_{-1}^{1}\frac{dx}{\sqrt{a^2+1-2ax}\sqrt{b^2+1-2bx}}$

show this integral $$I=\int_{-1}^{1}\dfrac{dx}{\sqrt{a^2+1-2ax}\sqrt{b^2+1-2bx}}=\dfrac{1}{\sqrt{ab}}\ln{\dfrac{1+\sqrt{ab}}{1-\sqrt{ab}}}$$ where $0<a,b<1$ my idea: let ...
1
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1answer
17 views

Check my answer - simple laplace transform of piecewise continuous function.

I'd just like to check that I got the idea right, first exercise im doing in laplace transforms and am a bit clueless. We are given $f(t)=0$ if $0<t<2$ and $f(t)=t$ if $t>2$. We are asked to ...
0
votes
1answer
49 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
2
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1answer
34 views

Why $ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $?

from this answer I could not see what is happening here: $$ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $$ What technique of integration ...
2
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1answer
43 views

How find this sum closed form $I=\sum_{k=1}^{n}\int_{0}^{+\infty}\cos{(2kx)}x^{m-1}e^{-ax}dx$

Find this closed form? $$I=\sum_{k=1}^{n}\int_{0}^{+\infty}\cos{(2kx)}x^{m-1}e^{-ax}dx,m\ge 1,a>0$$ use ...
4
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0answers
49 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
5
votes
1answer
74 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
1
vote
3answers
66 views

Seemingly Simple Integration: $x/(x-1)$

I am currently working on some advanced engineering math but this seemingly simple integral has me stuck. Someone please show me how to derive it. It is part of a far bigger more complex problem in ...
2
votes
2answers
25 views

Find the continuous function such that the Riemann integrable is the same

Find all functions $f$ such that $f$ is continuous on $[0,1]$ and $\int_0^x f(t) dt = \int_x^1 f(t) dt$ for every x $\in (0,1)$ I can't think of any function that would satisfy this property! ...
1
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1answer
43 views

What is the 'largest' space of integrable functions which is also a Hilbert space?

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that ...
4
votes
1answer
116 views

How evaluate the following hard integrals?

Prove: $$\displaystyle\int_0^{\frac{\pi}{4}}{\,x}{\,\arctan\sqrt{\frac{\cos2x}{2\sin^2x}}}dx=\frac{\pi}{96}[{\pi^2}-6\ln^22]$$ And ...
3
votes
2answers
75 views

Surface area of a solid of revolution: Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ work?

Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ yield the correct answer when calculating the surface area of a solid of revolution?
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0answers
34 views

Clarifying a step in an integration solution

In the accepted answer here, the first two steps in computing the integral are \begin{align} \mathcal{I} =&\frac{1}{2}\int^\infty_0\ln(1-e^{-2x})\ln\left(\frac{x^2}{\pi^2+x^2}\right)\ {\rm d}x\\ ...
0
votes
1answer
43 views

Integration problem that may use DCT

I am trying to solve the following problem. Let $f \in L^2(0,1)$ and define $$f_n(x)= n \int\limits_{k/n}^{(k+1)/n} f(y) dy $$ for $x \in [k/n, (k+1)/n)$, $k=0,1, \dots, n-1.$ Show that $f_n ...
1
vote
2answers
31 views

If $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure.

I want to prove that if $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure. This is the proof: Suppose not. Then there exist $\epsilon>0,\delta> 0$ such that $μ \{x: ...
1
vote
3answers
50 views

Fundamental Theorem of Calculus 1 - definite integral

I have two problems, they're not from a book so I can't check the answer for one of them and the other I'm not sure on what to do. $$ {d\over dx}{\int^{1}_{x^{2}}} {\sqrt{t^{2}+1}} {dt} $$ $$=-{d\over ...
0
votes
4answers
56 views

Expectation of non-negative random variable

Let $X$ be a non-negative random variable. In a proof for $E[X]=\int_0^\infty P(X>t)dt$ from the answer of this question, we use Fubini for the middle quality. Why do we need $X$ to be ...
0
votes
2answers
32 views

Bound on and integral

If $\alpha \in \Bbb R$, how can I show $$\int_{-M}^M \frac{1}{\sqrt{|x-\alpha|}} \, dx \le 4 \sqrt{M}$$ For $M>0$, Rewriting the integral gives $$\int_{-M+\alpha}^{M + \alpha} ...
1
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0answers
26 views

product of barycentric coordinates over a simplex

Show that: $$\int_K \eta_0^{\alpha_0}(x)\cdots\eta_n(x)^{\alpha_n}dx=\frac{{\alpha_0}!\cdots{\alpha_n}!n!|K|}{(\alpha_0+\cdots+\alpha_n+n)!}$$ where $\eta_i$ barycentric coordinates and $K$ is a ...
2
votes
3answers
28 views

Determine monotone intervals of a function

Let $$ f(x) = \int_1^{x^2} (x^2 - t) e^{-t^2}dt. $$ We need to determine monotone intervals of $f(x)$. I tried to differentiate $f(x)$ as follows. $$ f'(x) = \left(x^2 \int_1^{x^2} e^{-t^2}dt \right)' ...
1
vote
0answers
20 views

Finite integral involving branch cut. Basic Argument Question

I am reading this Wikipedia article on examples of contour integrals using complex analysis (http://en.wikipedia.org/wiki/Methods_of_contour_integration). In particular, I am looking at Example (VI), ...
2
votes
8answers
121 views

How to show that $f(x) = 0$ if $\int_a^bf(x)\,\text{d}x=0$ for all $a,b\in\mathbb{R}$?

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
3
votes
6answers
154 views

Proving that $\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$ converges with no trig functions

Let $$\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$$ How to show that it converges with no use of trigonometric functions? (trivially, it is the anti-derivative of $\sin ^{-1}$ and therfore can be ...
1
vote
0answers
17 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
1
vote
1answer
49 views

How to compute $\int^{1}_{-1}f(x)dx$?

I need to compute $\displaystyle\int^{1}_{-1}\,{\rm f}\left(\, x\,\right)\,{\rm d}x$, where $$ \,{\rm f}\left(\, x\,\right) =\left\{\begin{array}{lcrcl} x & \mbox{if} & x & \leq & 0 ...