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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1answer
5 views

volume of solid of rotation: finding r

For a region bounded by: $$y=x+4,\;y=16-x^2;\;around\;y=-5$$ I understand that I will be using the 'washer' method: $$V =\int_a^b\pi r^2h$$ But I'm having a hard time finding $$r^2 \text{ for}= ...
0
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1answer
18 views

Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
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1answer
14 views

Change of Variable involve derivative

Let me just give the 1-D version of my problem. Let $u\in C_c^\infty(R)$ and define $u_r(x):=u(rx)$. Then I am trying to evaluate the integration $\int_R u_r'(x)dx$. Here is my steps: $$\int_R ...
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votes
2answers
63 views

How would you solve $\int \frac{x}{x^2 - 4x + 5} dx$

What is the tip for integrating that integral? I completed the square on the bottom to make it $$\frac{x}{(x-2)^2 + 1}$$ but it doesn't seem helpful. Any tips? Thanks.
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2answers
28 views

Indefinite integral of fraction

I'm working through some indefinite integral exercises. There is one here that I can't seem to figure, and there is no solution in the textbook: $$\int \frac{3}{4x^2+4}dx$$ I'm assuming it has ...
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0answers
18 views

Question from integral with using fourier's integral

Please explain me how to compute this integral: $$ \int_0^\infty \dfrac{\cos(\omega x)+\omega \sin(\omega x)}{1+\omega^2}d\omega$$
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vote
1answer
23 views

Trigonometric Integration: Using the half-angle formula?

I'll preface my question by saying this is my first ever post. I've been lurking around and answering a couple logic questions here and there, but since I have an intractable calculus question I ...
1
vote
1answer
68 views

intregration without substitution of $x^x \ln x$

How do i integrate this without any substitution, purely algebraically : $$x^x \ln ex$$ I've tried a lot but not have been able to: $$x^x \ln (x + 1) = \ln x^{x^x} + x^x$$ or $e^{x \ln x}\ln ...
0
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1answer
16 views

How to describe two integration contours as set? [on hold]

Friends I need support to understand how one can describe two integration contours as set? can anyone please explain it with the help of a example?
2
votes
1answer
22 views

Finding $\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$ using Plancherel’s theorem

Suppose $G(\mathbb R)\ni f(x),\mathcal{F}[f](\omega)=\frac{1}{1+|w|^3}$ find the value of $$\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$$ I thought using Plancherel’s theorem \begin{align} ...
7
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1answer
69 views

A cute limit $\lim_{m\to\infty}\left(\left(\sum_{n=1}^{m}\frac{1}{n}\sum_{k=1}^{n-1}\frac{(-1)^k}{k}\right)+\log(2)H_m\right)$

I'm sure that for many of you this is a limit pretty easy to compute, but my concern here is a bit different, and I'd like to know if I can nicely compute it without using special functions. Do you ...
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0answers
27 views

What is integration contour and how to discribe it? [on hold]

We knew that an integration contour can be described as a set of points. How one can describe the two integration contours as sets.? Can anyone help me with examples.
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1answer
30 views

Integral using Beta Function and Gamma Function

Interestingly, I seem to have an integral I have posted before, but I want to take a different approach to it. $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} \,dx$ The beta function states, $B(x,y) = ...
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1answer
18 views

2D Fourier transfrom of $1/(x^2-y^2+q)$

How can I calculate the following 2D Fourier integral: $$ \iint \frac{{\rm e}^{{\rm i}(ax+by)}}{x^2-y^2+q} {\rm d}x\,{\rm d}y, $$ where $q$ is a complex number? If there was a "+" sign in the ...
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vote
4answers
81 views

Problem with a solution to the integral $\int_{-\infty}^{+\infty}e^{-x^2}\mathrm{dx}$

I am an undergrad in my first year of college. Today, our mathematics professor solved the integral $\int_{-\infty}^{+\infty}e^{-x^2}\mathrm{dx}$ which he called "One of the most important integrals ...
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2answers
53 views

Calculate $\int(1-\sin x)^2\cos x\,dx$ [on hold]

How to calculate the following integral? Calculate $\displaystyle\int(1-\sin x)^2\cos x\,dx$.
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2answers
68 views

Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$

Prove that $$\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$$ I tried to use Weierstrass substitution but the term $\cos 4x$ made horrible algebraic-forms since ...
1
vote
1answer
46 views

Trigonometric Integration.

Q. $$\int _0^{\frac{\pi }{4}}\:\left(\frac{1}{\left(\cos^4x-\cos^2x\sin^2x+\sin^4x\right)}\right)\:dx$$ My method: =>$$\int _0^{\frac{\pi ...
2
votes
0answers
18 views

Skew symmetric matrices even size commutativity

Given Data in the question $w(t)=\frac{1}{2}\begin{bmatrix} 0 &r(t) &-q(t) &p(t) \\ -r(t)& 0 &p(t) &q(t) \\ q(t)& -p(t) &0 & r(t)\\ -p(t)&-q(t) ...
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3answers
43 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
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0answers
19 views

How to find the volume of Solid Revolution

Can anyone help with this question, especially Part B? (a) Find the area of the region enclosed between the curves $y = x^2−2x+3$ and $y = x+1$. (b) The area described in the previous part is now ...
1
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1answer
23 views

Geometry of Riemann Stieltjes integration

We know that $\int_{a}^b f(x)dx$ represents the area bounded by the curve $y=f(x)$& the straight lines $x=a$ & $x=b$. But when we integrate $f(x)$ with respect to another function $g(x)$ then ...
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2answers
17 views

Integral of absolute value of X and area under the curve.

Here's my question. We know that the absolute value of X looks like: Clearly, we can see, since the absolute value of x is always greater than or equal to 0, the area under the curve is always ...
0
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1answer
56 views

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ prove the following:

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ such that $ f(0)\neq -1$ and $\displaystyle\int_{0}^{b} f(t) \, dt=0$ Show that the equation $\displaystyle\int_{x}^{a} f(t) ...
1
vote
1answer
31 views

Integrals using Parsevals Theorem

I've been assigned two integrals to calculate in Fourier Analysis: $$\int_{-\infty}^{\infty}\left(\frac{\sin x}{x}\right)^2dx$$ ...
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0answers
21 views

Integral Calculus: can we just add constants when calculating indefinite integrals?

Attached is a picture of a problem and its solution. My question is: why do we multiply the entire thing by 2 (second line one the left)? I know it helps get rid of the 2 in du, but where did it come ...
0
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0answers
20 views

Integral with a gamma functions inside

I have a function based on the binomial distribution, $$f(x;n,p)=\sum_{i=0}^{n} |x-i|\binom{n}{i} p^i (1-p)^{n-i}.$$ It's not so hard to plot this out with discrete points, but I'd like to smooth ...
0
votes
1answer
20 views

diffeomorphisms preserve zero measure

Suppose $\Omega\subset \mathbb R^N$ is an open set and $f:\Omega\rightarrow f(\Omega)$ is a $C^1$ diffeomorphism. Show that if $F \subset \Omega$ has zero measure then $f(F)$ has zero measure. I ...
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votes
3answers
151 views

Integral of greatest integer function divided by an exponential

If $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$, then find $\displaystyle\int_{0}^{\infty} \displaystyle \frac{\lfloor x \rfloor}{e^{x}} dx$. The correct answer is supposed to be ...
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0answers
18 views

How to calculate the integral $\int_{\mathbb R^n} e^{i\lambda |x|^\alpha + ix\cdot\xi} dx$?

I want to calculate the $n$-dimensional Fourier transform of the function $e^{i\lambda |x|^\alpha}$, where $\lambda\in\mathbb R$ and $\alpha \in \mathbb R$, that is, the value of the following ...
3
votes
1answer
47 views

How to evaluate the following integral? $\int\frac1{1+\sqrt{\tan x}}\mathrm dx.$

Evaluate the following integral: $$\int\dfrac1{1+\sqrt{\tan x}}\mathrm dx.$$ I know this question has a solution, but I haven't the slightest idea how to do it.
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2answers
16 views

Integral and making a substitution over a given area

If we have the integral $\int x^2 * e^{-x^2}dx$ in the area where $x>0$. Now if we make the substitution $y = x^2$, why does the integral then become $1/2\int y^{1/2} * e^{-y}$ as opposed to ...
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0answers
16 views

Continuous RV - minimizing absolute deviation

We try to find c value minimizing E[|x-c|], "expected value of absolute deviations", for a continuous random variable X. E[|x-c|]=Integral(-inf,inf)[|x-c|]f(x)dx ...
2
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1answer
46 views

help with strange Double Integral: $\iint_E {x\sin(y) \over y}\ \rm{dx\ dy}$

i'm having trouble with this double integral: $$ \iint_E {x\sin(y) \over y}\ \rm{dx\ dy},\ \ \ \ E=\Big\{(x,y) \in \mathbb{R^2} \mid 0<y\le x\ \ \ \land\ \ \ x^2+y^2 \le \pi y\Big\} $$ i've ...
0
votes
1answer
26 views

Integral by substitution does not work

I don't know where I am getting it wrong. I have this integral: $$ \int_{0}^{\infty}e^{-0.5t}\left(1-(1-f)e^{-zt}\right)dt $$ where $z>1$ and $f\in(0,1)$. I can solve directly this integral and it ...
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0answers
272 views

Inverting a Characteristic Function for half-cubic Student T entailing a Modified Bessel of 2nd kind

The Characteristic function of the Student T with $\alpha$ degrees of freedom, $C(t)=\frac{2^{1-\frac{\alpha }{2}} \alpha ^{\alpha /4} \left| t\right| ^{\alpha /2} K_{\frac{\alpha ...
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0answers
18 views

Integrating a particular function.

Could someone please show me how to integrate the following: $\int_{-\infty}^{\infty}p(x)L(p(x))dx$, where $L(x)$ is defined as $L(x)=\frac{x-1}{ln(x)}$ and $p(x)=(2\pi ...
2
votes
2answers
107 views

Integral difficulties (attempt included)

I am having difficulties with the following integral. I began working on it and thought I had obtained the answer, but when I went to graph it I received an integral of 1. I obtained the same answer ...
12
votes
4answers
170 views

How to proof the following function is always constant which satisfies $f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $?

Suppose that $f(x)$ is a bounded continuous function on $\mathbb{R}$,and that there exists a positive number $a$ such that $$f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $$ is constant. ...
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2answers
24 views

Weird question, antiderivative and integral

Given that $F(x)$ is the antiderivative to the function $f(x) = x^2 * ln(x)$ which satisfies that $F(1) = 7/9$, calculate $F(2)$. How does one do that? I first figured that I would calculate the ...
2
votes
3answers
55 views

Integration question: $\int \frac{\mathrm{d}x}{\sqrt{3 x} (3 x+1)}$

I am missing one piece of how to integrate the following: $\int \frac{\mathrm{d}x}{\sqrt{3 x} (3 x+1)}$ I found a solution to a similar problem which I entirely understand: I can usually use ...
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votes
2answers
15 views

Lower Sums and Partitions

Let $f\colon [0,1]\rightarrow \mathbb{R}$ be a bounded function. Let $P_1=[0,\frac{1}{2}]\cup [\frac{1}{2},1]$ and $P_2=[0,\frac{1}{3}]\cup [\frac{1}{3}, \frac{2}{3}]\cup [\frac{2}{3},1]$ be ...
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0answers
22 views

Solving Elliptic Integrals with variables

I have this complete elliptic integral of the second kind: $$ E\left(-\frac {4x}{(1+\delta)^2}\right) $$ I know I can solve the integral by giving a value to $\delta$ and $x$. However, I can't because ...
2
votes
3answers
111 views

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$ WolframAlpha gives a numerical answer of $43.8122$, which appears to be $\sqrt{611\pi}$. And playing with that, it seems ...
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1answer
20 views

Signed Measure Decomposition

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! A complex measure can be decomposed by exploiting the Radon-Nikodym derivative: ...
0
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1answer
25 views

Complex Measure Integration

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! Integration w.r.t. complex measure usually is defined via the Radon-Nikodym ...
0
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1answer
40 views

$\int e^{x^2}(y-1) \,dx$ [on hold]

What is $\int e^{x^2}(y-1) \,dx$ ? I could not find the answer.
0
votes
1answer
32 views

Work required to pump water out of a tank in the shape of a right circular cone

A tank in the shape of a right circular cone is full of water. The tank is 6ft. across the top and 8 ft. high. How much work is done in pumping water over the top edge. (a) Set up the integral (b) ...
4
votes
3answers
185 views

Integrate by partial fraction decomposition

$$\int\frac{5x^2+9x+16}{(x+1)(x^2+2x+5)}dx$$ Here's what I have so far... $$\frac{5x^2+9x+16}{(x+1)(x^2+2x+5)} = \frac{\mathrm A}{x+1}+\frac{\mathrm Bx+\mathrm C}{x^2+2x+5}\\$$ $$5x^2 + 9x + 16 = ...
0
votes
2answers
26 views

Trigonometric Substitution Evaluate ∫(x^3)/(sqrt(9-4x^2))dx

Evaluate ∫(x^3)/(sqrt(9-4x^2))dx. i started by making the bottom sqrt((3)^2-(2x)^2)) and let 2x=3sinu so, 2dx= 3cosudu => dx=(3/2)*cosudu and then i plugged the new values in and got ∫(((3/2)sinu)^3 ...