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

rotation n the coordinate space x, y, z

Let $R = \{\ (x,y) : x \geq 1, 0 \leq y \leq \frac{1}{x} \}$. a) Prove that $R$ has finite area. b) Prove that $R$ when rotate the coordinate space x, y, z about the x axis we get a solid of finite ...
0
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0answers
18 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
2
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0answers
20 views

Taylor expansion of a not easily differentiable function

Context: I'm trying to find the period of a simple pendulum. As is well known, if the initial angle is small the period is approximately constant. I'm trying to do a second order expansion. I have ...
2
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1answer
34 views

The $\frac{1}{x+i\varepsilon}$ distribution.

I read that the distribution defined as: $$ \lim_{\varepsilon \rightarrow 0}\frac{1}{x+i\varepsilon}$$ is equal to $$p.v. \frac{1}{x} -i\pi \delta(x)$$ So that for any test function $f$, ...
0
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0answers
8 views

Integrating along a line of point sources

I have some concentration that radiates from a spherical point, being steadily consumed until it hits zero at some distance $r_{n}$. This is given by $$C(r) = A\left[r^2 + \frac{2r_{n}^3}{r} - ...
2
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0answers
13 views

Simplifying a Fourier integral

I have what is effectively a Fourier integral resulting from applying Abbe's theorem that I would like to simplify (ideally into a closed form solution): $$ \int_{\theta_1}^{\theta_2} \alpha \beta ...
3
votes
0answers
65 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
0
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1answer
28 views

Proving integrability of a function in a domain, and solving another.

Please help if you can, thank you very much. I need to prove the integrability of the following integral: $$\int_0^{\pi/2}\frac{x \sin (\tan(x))}{\cos(x)}dx$$ I tried to prove it is bounded and ...
2
votes
3answers
82 views

Find the particular solution of the equation that satisfies condition

Full question: Find the particular solution of the equation $f'(x) = 4x^{-1/2}$ that satisfies the condition $f(1) = 12$. I have $f'(x) = 4x^{-1/2}$ and am given that $f(1) = 12$. I took the ...
2
votes
1answer
40 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
1
vote
1answer
51 views

Why does $\int_1^\infty\frac{\sin^2(x)}{x}\mathrm d x$ diverge?

Why does $\displaystyle\int_1^\infty\dfrac{\sin^2(x)}{x}$ diverge ? Why does Dirichlet-Test not work ? Define $f(x)=g(x)=\dfrac{\sin(x)}{x^{1/2}}$, Then $\forall b>1$ and $a>0;$ ...
1
vote
1answer
21 views

Finding the integral for function with floor

I am trying to find the integral $$\int_0^n f(x)dx$$ where $$f(x) = 2ax - {1 \over 2}a^2 - {1 \over 2} a $$ and $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + \sqrt 5 ...
0
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4answers
50 views

Integration Help: $\sqrt{(x^4 + x^2)}$

How do you integrate $\sqrt{(x^4 + x^2)}$?
2
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1answer
13 views

Getting from a product of gamma functions to a fraction answer

I am working on an assignment question for my Advanced Calculus course and am having great difficulty working it out. In order to try and understand this type of question/working, I have found a ...
3
votes
0answers
35 views

$\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)$

Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5),\quad H_n:=\sum_{k=1}^n\frac{1}{k}\ \ ...
2
votes
2answers
44 views

Problem calculating line integral

I have $\gamma=[0,1]\to\mathbb{R}^3$ defined by $\gamma(t)=(\cos(2\pi t), \sin (2\pi t), t^2-t)\;\forall t\in[0,1]$ and I'm asked to calculate ...
-1
votes
2answers
42 views

Antiderivative of $e^{au}$

I cannot seem to figure out $\int e^{au}du$. I have tried u-substitution (of course with a variable other than u) and can't get it to work out to the right answer. Any help is appreciated. Thanks!
0
votes
0answers
14 views

Simplifying Inequalities Before Converting Cartesian Coords. to Polar

I have a 3 dimensional region defined by Cartesian coordinates and I have to convert them to cylindrical coordinates. That is the easy part, but what I don't understand is how to treat the ...
2
votes
0answers
33 views

What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
0
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0answers
60 views

How to solve this integral ($\int _{\frac{\pi }{6}}^{\frac{\pi }{4}}\sqrt{1-\tan ^2\left(x\right)}dx$)

$$\int _{\pi/6}^{\pi/4}\sqrt{1-\tan ^2\left(x\right)}dx$$ Hey, can you help me to solve this integral please? Thanks.
1
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1answer
43 views

Volume of the solid bounded by the planes (Checking the limits of the integral)

Find the volume $V$ of the solid bounded by the planes $x+y-z=3$ and $z=0$, and the cylinder $x^2+\frac{y^2}{4}=1$. My calculations give Polar $$V = \int_{\theta=0}^{\theta=\pi/2} \int_{r=0}^{r=1} ...
0
votes
1answer
33 views

Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
2
votes
2answers
26 views

Evaluating expression at infinity

How do I evaluate something like: $$xe^{-(x-\theta)}\text{ from }x = \theta\text{ to }x=\infty?$$ This came up in an integration I tried to do, and I realize it's a very basic question. But I am ...
1
vote
1answer
71 views

Evaluating $\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$ using polar coordinates?

How is the following integral found using polar coordinates. $$\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$$ I know the the part of the domain the circle being asked in ...
0
votes
1answer
41 views

Don´t know how to start proving this formula.

\begin{equation*} \int \frac{\cos ^{m}x}{\sin ^{n}x}dx=-\frac{\cos ^{m+1}x}{(n-1)\sin ^{n-1}x}- \frac{m-n+2}{n-1}\int \frac{\cos ^{m}x}{\sin ^{n-2}x}dx+C,\qquad (n\neq 1). \end{equation*} I`d like to ...
0
votes
2answers
41 views

Evaluate an integral using polar $\displaystyle\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\,dy\,dx$

How do you evaluate the following integral using polar cordinates. $$\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\:\mathrm{d}y\:\mathrm{d}x$$ I converted it to polar coordinate making it ...
1
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3answers
46 views

Are discontinuous functions integrable? And integral of every continuous function continuous?

According to me answer of second part is yes as integration simply means area under curve.
2
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1answer
33 views

how ro prove f(x,y) is integrable in $[a,b]\times[c,d]$

If there exits a $f(x,y)$ in $\mathbb{R}^2$,and if we fix any $x$ in $[a,b]$, then $f(x,y)$ is increasing as $y$ increases. Also, if we fix any $y$ in $[c,d]$,the $f(x,y)$ is increasing as $x$ ...
2
votes
2answers
34 views

Integration by reduction

I have learnt how to integrate by reduction formula but this one seems to give me hell someone to lift me by telling me what to do or simply to solve it. \begin{equation} I_n=\int\sec^n x\,dx ...
1
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0answers
36 views

Computing the values of $ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{2n} $

Define the following integral as $$ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{n}\,,\quad V_\alpha(n):=\int_0^\pi x^{\alpha-1}\cos(x)^{n} $$ where $n \in\mathbb{N}$. Now in the base case ...
3
votes
0answers
28 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
0
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1answer
16 views

Question about Riemann Integration and the indicator function

Let $S \subseteq \mathbb{R}^n$. Suppose $\chi_S$ is integrable and $\int_Q \chi_S = 1 $ for some rectangle $Q$ such that $S \subseteq Q $. Let $\epsilon > 0 $ be given, I want to ask how can I ...
0
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1answer
42 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
2
votes
1answer
51 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
0
votes
1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
0
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1answer
38 views

Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
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1answer
34 views

Inverse integration of a special definite integral

Hi I am facing a problem with this problem. please help.
2
votes
2answers
105 views

Why do we bother with $u$-substitution?

This question has bothered me ever since I learned $u$-substitution (A note here: I have no formal education at this level, so I may definitely have missed something). The method is presented as an ...
0
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2answers
24 views

How to integrate by reduction method

How to evaluate the integrals of (a) $(\ln(x))^n$ (b) $x^ne^{ax}$ where $a$ is a constant By reduction formula
2
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1answer
30 views

Re-interpreting double integral as a Type II Region $\mathrm{d}y\,\mathrm{d}x$ vs $\mathrm{d}x\,\mathrm{d}y$

I have the following Double Integral:$\iint_Dx\cos y\space\mathrm{d}A$ where $a$ is bounded by $x=1,y=0,y=x^2$. Interpreting this region as a Type one region, it is easy to conclude $R=\{(x,y)\mid ...
0
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3answers
43 views

How to solve integration of $\int x(x^2+k^2)^{-1/2} \, dx$?

As said in title, how do you solve integral $\int x(x^2+k^2)^{-1/2}\,dx$ where $k$ is some constant?
6
votes
2answers
118 views

Compute $I=\int_0^{+\infty}\frac{\arctan(t)}{e^{\pi t}-1}dt$

I would like to compute $\displaystyle I=\int_0^{+\infty}\frac{\arctan(t)}{e^{\pi t}-1}dt$ Let $D=(0,+\infty)$, I have $\frac{1}{e^{-\pi t}-1}=\frac{e^{-\pi t}}{1-e^{-\pi t}}$ So ...
1
vote
1answer
36 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
1
vote
1answer
46 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
3
votes
3answers
63 views

Convergence of $\int_{0}^{+\infty}\ln(1+\frac{1}{t^2})$

Study the convergence of $\int_{0}^{+\infty}\ln(1+\frac{1}{t^2})dt$ For $+\infty$ case it's easy we have $\ln(1+\frac{1}{t^2})\sim \frac{1}{t^2}$ For $0$ case I feel it's ...
0
votes
1answer
43 views

Are there integrals you can't solve without inverse hyperbolic substitution?

Are there any integrals that can't be solved with only trig substitution? An integral that requires you to use a hyperbolic or inverse hyperbolic substitution?
1
vote
0answers
26 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
2
votes
0answers
27 views

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
1
vote
2answers
87 views

Evaluate $\iiint xyz$

Evaluate $$\iiint_E xyz\, dV$$ where $E$ is the solid: $0\leq z\leq 9,\,0\leq y\leq z,\, 0\leq x \leq y.$ I am having a hard time drawing a picture of this solid $E$ to find out what the ...
2
votes
0answers
34 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...