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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15 views

Liouville's Extension of Dirichlet Theorem

Can we use Liouville's Extension of Dirichlet Theorem to find triple integral $\int\int\int(u^2+v^2+w^2)\space du\space dv\space dw\space where\space u=0, v=0, w=0\space \&\space u+v+w\leq1$? Or ...
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1answer
27 views

Integration without substitution

How to i integrate this with out substitutions or Partial fraction decomposition ? ($3$$x^2$+$2$)/[$x^6$($x^2$+1)] I've got to : 2/x^6(x^2+1),but after this i haven't been able to eliminate the 2.
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0answers
4 views

Finding an optimal path for minimizing an integral.

Let $f(x,y)$ be real-analytic in both $x$ and $y$. How the find a path from (0,0) to (1,1) such that the integral over that path has the minimum absolute value ?
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4answers
39 views

Definite Integrations problems [on hold]

If $f(x)= x^2 e^{x^2}$ then show that $f'(x)= 2xe^{x^2} + 2x^3 e^{x^2}$ and use this result to evaluate $$\int x^3 e^{x^2} \, dx$$ How can I use the result to evaluate the integral?
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2answers
16 views

Behaviour of the error function as $z \rightarrow -\infty$?

I'm trying to find the behaviour of the error function, $erf(z)$ as $z \rightarrow -\infty$ $$erf(z) = \frac{2}{\sqrt{\pi}}\int_0^{z} e^{-s^2}\mathrm{ds}$$ I know that we can find the limit of ...
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0answers
8 views

A question about integral-squared error.

We consider the problem of representing a time function, or signal, $x(t)$ on a $T$-s interval $(t_0, t_0+T)$, as an expansion. Thus we consider a set of time functions $\phi_1 (t), \phi_2(t), ..., ...
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2answers
145 views

Simplest way to integrate this trigonometric integral:

$$\int \frac{1}{1+\tan x}dx,$$ A substitution like $t = \tan x, \;dt = (1+t^2)dx$ etc. immediately comes to mind, but I find this method a bit lengthy with the partial fractions. Is there a more ...
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7answers
224 views

Proving area under the integrals.

I have a question that I have been trying to solve that I am curious about. If you have a continuous function $f(x) = \frac1x$. How would you prove that ...
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0answers
13 views

Showing equality between integral and shifted Fourier series

Let $f\in E[-\pi,\pi]$ and let $f\sim \frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx +b_n\sin nx$ be the fourier series of f in $[-\pi,\pi]$.show that $$\forall -\pi\le c,x\le \pi \quad ...
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0answers
18 views

What is the domain of the inverse function

$f(x)=(x + 6)/(4x + 3)$ (a) Find the inverse function of $f$. My solution to (a) is $x=(y+6)/(4y+3)$ $(y+6)/(4y+3)=x$ $y+6=4xy+3x$ $y+6-4xy=3x$ $-4xy+y+6=3x$ $y=-(3(x-2))/(4x-1)$ I replaced ...
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3answers
39 views

Evaluating $ \int {e^x \sin (k \pi x) } dx $

I'm trying to integrate $$ I = \int {e^x \sin (k \pi x)} dx. $$ I've used Matlab and Wolfram Alpha, which have both given me the result $$ I = \frac{e^x(\sin (k \pi x) - \cos (k \pi x))}{k^2 \pi^2 +1 ...
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4answers
30 views

Taking partial fractions for integration?

I'm having some trouble with integrals involving partial fractions it seems. Been stuck on this forever. The equation is given below and I have to use partial fractions to solve. ...
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0answers
10 views

Hypothesis testing CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
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0answers
11 views

Solve $\int\limits_{\mathbb{R}^n} e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$

How to calculate $$\int_{\mathbb{R}^n}e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$$ where $\langle \cdot, \cdot \rangle$ denotes the canonical inner product in $\mathbb{R}^n$. I'm trying use ...
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0answers
19 views

justification of step in complex integration

What is the justification for the step with the red square next to it, how do we change the integrator like this?
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1answer
26 views

Tough function to integrate

I am having trouble seeing the process to integrate this function (wrt T) $$A(T) = \frac{1}{(CT^4)}\frac{(1-a)K+aT}{a(K-T)}$$ Should I use integration by parts? I do not see how this will work
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1answer
18 views

Solve $\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx $

How to prove that $$\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx = 0,$$ where $c>0$?
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1answer
13 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}= ...
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1answer
19 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
16 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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2answers
65 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
29 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
21 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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1answer
28 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 ...
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1answer
72 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 ...
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1answer
17 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
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1answer
26 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} ...
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1answer
81 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
29 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
38 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
30 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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4answers
86 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
55 views

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

How to calculate the following integral? Calculate $\displaystyle\int(1-\sin x)^2\cos x\,dx$.
3
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2answers
75 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
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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
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0answers
19 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
44 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
20 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 ...
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1answer
57 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
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1answer
33 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 ...
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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
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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 ...
5
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3answers
154 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
20 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 ...
1
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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 ...