Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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1answer
51 views

How can I compute this integral $\int \cos^{2}\left(t\sqrt{x^{2}-1}\right)dx $

How can I compute this integral $$\int \cos^{2}\left(t\sqrt{x^{2}-1}\right)dx $$ Even when I use Maxima, it does not give result. Thank you very much.
2
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2answers
56 views

Suppose $\int_0^9f(t)dt=12$. Then is it true that $\int_0^3f(3x)dx = 12$

Suppose $\int_0^9f(t)dt=12$. Then is it true that $\int_0^3f(3x)dx = 12$ ? How do I go about figuring this out? I tried differentiating and using fundamental theorem of calculus but couldn't figure ...
0
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0answers
19 views

Convergence behaviour of Eichler integral

Consiger $g : \mathbb H \to \mathbb C$ a modular form of weight $2-k, k \in \frac{1}{2}\mathbb Z$. Let $z \in \mathbb H$ and consider the following integral: ...
0
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0answers
10 views

Line integral over a vector field

Evaluate ∫C < −y, x − 1 > dr where C is the closed piecewise continuous curve formed by the line segment joining the point A(− √ 2, √ 2) to the point B( √ 2, − √ 2) followed by the arch of the ...
0
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3answers
40 views

For what values of K, is the integral improper?

For what values of $K$ ($K > 0$), is the following integral improper? $$\int_{0}^{K}\frac{x}{x^2-2}$$ Now, I know that the function is undefined at $x=\sqrt{2}$. I also figured out that the ...
1
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1answer
29 views

How is the following integral rigorously meant to be understood?

Consider $\mathbb{R}^3$. Consider the following integral on the unit three sphere $$ \int_{S^3}\frac{1}{x^2}\,d^3x $$ where $x^2=x_1^2+x_2^2+x_3^2$. I have quite some working knowledge on integrals ...
0
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1answer
34 views

Simpson's rule is not producing better results than Riemann sums

I have to calculate RMS value $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ and I know from the maths that the Simpson's rule should provide better approximation of the definite integral than the Riemann sums. ...
0
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2answers
115 views

Anti-Derivative of $\ln(x^2 + 7)$ is kicking my butt, can anyone help?

I'm given $\ln(x^2 + 7)$ in a problem and to solve it I need to get the anti-derivative, but I haven't been able to properly calculate it. Could someone show me how to obtain this anti-derivative? It ...
3
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1answer
50 views

Antiderivative of $\arctan(-x^2)$

As I said in the title I'm trying to find an antiderivative of $$f(x)=\arctan(-x^2)$$ I am aware that e.g. WolframAlpha can find one, but I have no clue how to do it by hand. Can anyone give me a ...
0
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1answer
34 views

Using divergence theorem to calculate surface of sphere

I want to calculate: $$\iiint_V div (\overrightarrow F \cdot \space dV) $$ with $\overrightarrow F=x^3\hat i+ y^3 \hat j+z^3\hat k$ and Surface of sphere given as $x^2+y^2+z^2=r^2$ So, first I ...
1
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0answers
28 views

Newtons Law Of Cooling (And Heating)

Rule is: $D= A.e^{-kt}$, Where: $k,a$ are elements of real numbers, $D$ is the difference between the temperature of the item and the surrounding air, and t is the time in hours since the object ...
1
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2answers
73 views

Evaluate $\int \frac{\sqrt{x}+1}{\left(\sqrt[3]{x}-1\right)\sqrt[6]{x^5}}\,dx$

How should I approach this? $$\int \frac{\sqrt{x}+1}{\left(\sqrt[3]{x}-1\right)\sqrt[6]{x^5}}\,dx$$ I intend to continue with the substitution method, however, I find it difficult to understand what ...
1
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1answer
21 views

Double integration of a function of e to y squared where one integral has a variable

I'm not sure how to phrase the title, but I have a problem on my homework assignment that requires me to solve the following function. $\int_0^2 \int_{x^2}^4 xe^{y^2} dy dx$ WolframAlpha gives a ...
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0answers
8 views

Approximating the integral of a large product

I would like to approximate the following integral of a product: $$ I = \int dz\, f(z)\prod_{i=1}^n\left(1 - \rho_i(z)\right) $$ The functions $f$ and $\rho_i$ are differentiable for all $i$, ...
0
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1answer
25 views

Relation between $\lim_{n \to \infty}\int_{I}f_n(x)\:dx$ and $\int_{I}\lim_{n \to \infty}f(x)\:dx$ [on hold]

Relation between convergence and integration of sequence of a function. Let $f_n$ be a sequence of integrable functions defined on an closed interval with $$f_n(x) \to 0$$ on this interval ...
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3answers
63 views

Contour Integral of $\int\limits_0^{2\pi}\frac{d\theta}{1+a\cos\theta}$ for $a^2<1$ (textbook wrong?)

My book is telling me that the answer is $\frac{2\pi}{\sqrt{1-a^2}}$. I'm getting an extra a on the numerator. Could somebody verify if I'm wrong, or if it's my book (it has been wrong numerous ...
0
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1answer
21 views

Transforming an integral to a different domain

For a given $v(x)$ with $x\in[0,1]$, use the variable transformation $x=g(\eta)=\frac{1}{2}\eta+\frac{1}{2}$ to transform the integral $I=\int_0^1v(x)dx$ to an integral over $[-1,1]$. My doubts: ...
1
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3answers
46 views

Solve $\int_{0}^{1}y(xt)dt=ny(x)$ [on hold]

Solve $\int_{0}^{1}y(xt)dt=ny(x)$ Could someone help me to solve this?
0
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1answer
64 views

Find the mass of the half circle

Find the mass of the half circle that is defined by $x^{2} + y^{2} \le 4$ $(y \le 0)$ if the density at point $(x,y)$ is proportional to its squared distance from the point $(0, -2)$ and the density ...
0
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0answers
10 views

Line integral of vector field with singularity

Supose a vector field $\vec{F}(x,y)$ has a singularity at $(x_0,y_0)$. If I wanted to evaluate the integral $$\int_{\gamma}\vec{F}.d \vec{r}$$ along $\gamma$, knowing that $(x_0, y_0) \in \gamma$, how ...
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1answer
28 views

Integration/Functions Question [on hold]

I'm having trouble with this question, especially the integration in parts (f) and (g). Could someone please solve and explain? Thanks!
3
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0answers
32 views

Example of $f:]-1,1[ \rightarrow \mathbb{R}$ such that $G(x)=\int_0^x f(t)dt$ is not $f$'s primitive?

I cannot figure out the following: Give an example of integrable $f:]-1,1[ \rightarrow \mathbb{R}$ such that $G(x)=\int_0^x f(t)dt$ is not $f$'s primitive?
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1answer
23 views

Suppose f′′ continuous in [a,b] check:

I am solving some exercises of calculation book, and I'm stuck in this exercise, and I do not know how to make this exercise.... Suppose $f''$ continuous in $[a,b]$ check: $$ f(b) = f(a) + ...
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votes
2answers
30 views

Moments of a continuous random variable (Exercise 4.3.3 from Grimmett and Stirzaker)

Let $X$ be a non-negative random variable with density function $f$. Show that $$ E(X^r) = \int_0^\infty r x^{r-1} P(X > r)\,dx. $$ I tried using integration by parts to obtain \begin{align} ...
0
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1answer
29 views

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$. I'm thinking about a contradiction proof. Assuming ...
1
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1answer
30 views

Integral with Dirac delta function

We are given that: $$u(x,t)=\frac{2}{\pi} \int_0^\infty e^{-k^2t}G_s(k)\sin {kx}\space\text{d}k,$$ where $G_s(k)$ is the Fourier sine transform of $g(x)$. Find the solution $u(x,t)$ when ...
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1answer
46 views

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? [on hold]

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? Thank you very much.
1
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1answer
35 views

Area between two curves (Demidovich)

I'm trying to solve some problems on definite integrals from Demidovich's book and I'm stuck on calculating the area between two curves defined by: $$y_1 = \frac{a^3}{a^2 + x^2}, y_2 = 0$$ Any hints ...
0
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0answers
31 views

How to solve this PIDE?

How to solve for this partial integro-differential equation? with: as constants and where are constants Thank you very much
0
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1answer
44 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx ...
0
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0answers
14 views

Line integral of $F = \langle xz, xy , 3xz\rangle $ [on hold]

Let $F = \langle xz, xy , 3xz\rangle $ be a vector field. Let $c$ be the boundary of the plane $2x + y+z =2$ in the $1$st octave, counterclockwise from above. Then how can I compute the line integral ...
0
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1answer
50 views

Riemman Sum proof: x^2 [on hold]

How can I prove that $$\int _{a}^{b} x^2 = \frac{b^3-a^3}{3}$$ Using the definition of Riemman Sum
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0answers
29 views

Having trouble understanding what exactly I am integrating for a hydrostatic force/pressure calculus question

So the question is: a trough has vertical ends that are trapezoids with parallel sides of length 4m (top) and 2m (bottom) and a height of 3m. If the trough is filled with water to a depth of 2m, find ...
2
votes
1answer
48 views

Calculate this Triple integral!

They ask me find the following: W is the solid bounded by the limited right circular cylinder: $$ x^2+y^2=1$$ and the planes: $$z=0, z=4$$ must calculate: $$\iiint_W z\frac{e^{2x^2+2y^2}}{2} ...
5
votes
1answer
88 views

Is my proof correct? If $f$ has a finite number of discontinuities on $[a, b]$, then it is integrable on $[a, b]$

Question: Suppose a function $f(x)$ over the interval $[a, b]$ is bounded and has only a finite number of discontinuous points on $[a, b]$. I intend to prove that it must be integrable on $[a, b]$. ...
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1answer
24 views

Forms of functions in dynamical systems

I wanted to read some introductory material about dynamical systems since I might need a basic understanding of them in a related task. So, as far as I see, in a continuous time dynamical system, we ...
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0answers
13 views

Evaluating definite integrals with monomials tacked on the evaluation

Say we have the integral $$\int_B^L x \sin(ax) = - \frac{x\cos(ax)}{a}\big|_B^L + a^{-2}\sin(ax)\big|_B^L$$ And we want to evaluate the first cos term, do we do it like so: ...
1
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1answer
31 views

Use Plancherel theorem to prove the identity

Plancherel theorem states $\int \hat f(x) \hat g(x) \, dx=\int f(x)g(x) \, dx$, where $\hat f$ denotes the Fourier transformation of $f$. It is required to prove this: ...
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0answers
21 views

Spatial derivative of pendulum to time derivative

$m\ddot{x}\dot{x}+\frac{dV}{dx}\dot{x}=0\Rightarrow \frac{d}{dt}\left [ \frac{1}{2}m\dot{x}^{2}+V\left ( x \right ) \right ]=0$ $Using \frac{dV\left ( x\left ( t \right ) \right ...
6
votes
2answers
90 views

Evaluation of $\displaystyle \int_{0}^{\pi}\ln(5-4\cos x)dx$

Evaluation of $\displaystyle \int_{0}^{\pi}\ln(5-4\cos x)dx = \int_{0}^{\pi}\ln(5+4\cos x)dx$ $\bf{My\; Try::}$ Let $\displaystyle I(a,b) = \int_{0}^{\pi}\ln(a+b\cos x)dx$ Then ...
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2answers
77 views

Find “$g(x)$” knowing that “$x=\int_{0}^{\infty} g(tx) dt$”???

The entire question states what I am looking for. I'm looking for a function $g(x)$ in terms of $x$ which satisfies the condition that follows. This seems like it's related to "integral ...
0
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2answers
10 views

Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$

Suppose that $F(s)=\mathcal{L}\{f(t)\}$ (Defined as the usual Laplacian operator). Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$. ...
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2answers
40 views

Check that for every natural $n \ge 1$, has:

I am studying integrals by parts, and in the book I'm studying has an exercise in integrals by parts chapter, I do not even know where to start and I do not know how to proceed ... The exercise says: ...
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1answer
58 views

Find the triple integral $\iiint_{\mathrm{T}}x^{4}\, dxdydz$

Find the triple integral $\iiint_{\mathrm{T}}x^{4}\, dxdydz$ where $T$ is bounded by $x^{2} + y^{2}+2y = 0$ and $z = 2x, z = 0$ planes. My attempt at the solution: $0\le z\le 2x, -2\le y\le 0, ...
1
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2answers
46 views

Finding an integral involving logarithmic functions: $\int_0^\infty\frac{1}{z[\ln(z)]^2}dz$ [closed]

Finding the following integral: $$\int_0^\infty \frac{1}{z[\ln(z)]^2} dz$$
1
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1answer
15 views

Boundary change in double integral: $\int_t^T \left( \int_t^u r(s)ds \right)du = \frac{1}{T}\int_t^T (T-s)r(s)ds$

I have following problem. I have been reading an article on pricing Asian options and I have found one article directly concerning my topic. However, it is horribly written and I am trying to ...
0
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0answers
21 views

symmetry of two IID random variables [duplicate]

Suppose that $X$ and $Y$ are independent and identically distributed. The claim is that $P(X<Y)=P(X>Y)=1/2$. How do I prove this? My attempt Since they are IID $f_X=f_Y$. So ...
1
vote
0answers
64 views

How do I prove the following equation involving $\int_{0}^{+\infty}\left(\frac{sin(x)}{x}\right)^ndx$

$\int_{0}^{+\infty}\left(\frac{sin(x)}{x}\right)^ndx=n\int_{0}^{+\infty}\frac{x^{n-2}}{(x^2+2^2)(x^2+4^2)...(x^2+n^2)}$ if n is even and ...
1
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2answers
62 views

Integrating functions I've never dealth with

If $a>0$ show that $$\lim_{n \to \infty} \int_a^\pi \frac{\sin(nx)}{nx} dx = 0.$$ I've never dealt with non-elementary integral functions before and I'm not sure why this would show up on a ...