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
22 views

Numerically integration with a an infinite upper limit and non-zero lower limit

I have seen lots of quadrature formulas where we have definite limits or one of the limits is infinity and the other is zero. But what about the following case $$f(x) = \int_a^\infty e^{\frac{x}{t}} ...
1
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1answer
29 views

Integration of $\frac{dx}{(6x-4x^2)^{1/2}}$ and completing square.

Integrate $$\frac{dx}{\sqrt{6x-4x^2}}$$ While completing the square of $6x-4x^2$, I want to know where did $9/16$ come from in the following after taking $4$ out as the common factor $$-4\bigg(x^2 ...
0
votes
0answers
24 views

Laplace transform of a definite integral

I'm having some troubles with what follows. I am interested in finding the Laplace transform w.r.t. $x$ of some real-valued, positive, continuous (in general well-behaved) function $f(x,t),x,t>0$. ...
0
votes
1answer
35 views

Do we have $\int_{[0,1]^3} \min (x,y,z)\,dx\,dy\,dz=6\int_0^1\int_0^y \int_0^z x \,dz\,dy\,dx?$

I have this nice question: $\int_{[0,1]^3} \min (x,y,z)\,dx\,dy\,dz .$ I think it equals $6\int_0^1\int_0^z \int_0^y x \,dx\,dy\,dz .$ Then using my formula I got the result to be $\frac {1}{4} $ ...
0
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1answer
57 views

Is $\int_{|z|=2}\frac{z}{(z-3)^2}dz=0?$

I have a question. What is $$\int_{|z|=2}\frac{z}{(z-3)^2}dz?$$ In my optinion it must be zero, because the singularity $3$ is outside $\{z\in\mathbb{C}:|z|<2\}$, is it correct? Regards
1
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3answers
88 views

How to solve $\int \frac{1}{1-y^2}$ with respect to $y$?

I was solving an A Level paper when I came across this question. I tried substitution, but I'm not getting the answer with that. Would appreciate it if someone would help me.
0
votes
1answer
23 views

Evaluate [$-x^{n+1} e^{-x} ]_{x=0}^{x=\infty}$

Evaluate the following: [$-x^{n+1} e^{-x} ]_{x=0}^{x=\infty}$, where $n$ is any integer greater than 1 Any help?
-4
votes
2answers
78 views

Prove $\int x^n\,dx=\frac{x^{n+1}}{n+1}.$ [closed]

How to prove $\int x^n\,dx =\frac{x^{n+1}}{n+1}.$ I know integration is area under curve but how to continue any ideas.
0
votes
5answers
75 views

Explain solution to $\int\frac{ 7\,dx}{x(x^4 + 2)}$.

Can someone please explain how they get from the step outlined in red to the one in blue? I tried using partial fractions to break it up but that didn't work, and I'm not sure how else I could ...
0
votes
1answer
47 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \to ...
1
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2answers
63 views

What are the lightest hypothesis needed to be able to get the limit inside the integral?

Let $\{f_n\}$ be a sequence of Riemann integrable functions. What are the lightest conditions on $f_n$ to guarantee the following? $$ \lim_n \int_a^b f_n\,\mathrm dx=\int_a^b \lim_n f_n \, \mathrm ...
0
votes
4answers
63 views

Using trig identities to evaluate $\int_{0}^{\pi/2} \sqrt{1-\sin x} \, dx$

Use the identities $$\cos 2x=2\cos^2 x -1=1-2\sin^2 x$$ $$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$ to help evaluate $$\int_{0}^{\pi/2} \sqrt{1-\sin x} \; dx$$ I've already done ...
1
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1answer
27 views

Bump functions converging to an indicator

Suppose $K\subset\mathbb{R}^n$ has a smooth boundary, and let $\phi_s(x)$ be bump functions converging pointwise to the indicator of $K$, i.e. ...
0
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0answers
54 views

Surface Integral over Ellipse

Let $$E=\left\{(x_1,x_2)~\middle|~\frac{x_1^2}{ a^2}+\frac{x_2^2}{b^2}=1\right\}=\left\{X(\theta)=(a\cos(\theta),b\sin(\theta))\,\middle|\,0\leq \theta\leq 2\pi\right\},$$ be an ellipse. Let ...
1
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0answers
27 views

Integral in vector form

I have two vectors $x_{1}$ and $x_{2}$ and some function $f(x, k)$. So for example function $f$ can be evaluated at some point say $f(x_{1}, 10)$. Then I have an integral written ...
1
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0answers
15 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
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votes
2answers
94 views

Solve $\min_{A \subset \mathbb{R}} \int_{A} (f(t)-g(t))dt$

Consider the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ both integrable on any measurable set $A \subset \mathbb{R}$. Consider $$\min_{A \subset ...
0
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0answers
73 views

Is there a way to evaluate this integral? What are the bounds on its solution set?

I came across this integral in a problem I was trying to solve yesterday: $$ ...
1
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1answer
38 views
+50

Complicated surface integral/line integral.

Problem Compute the integrals $$I=\iint_\Sigma \nabla\times\mathbf F\cdot d\,\bf\Sigma$$ And $$J=\oint_{\partial\Sigma}\mathbf F\cdot d\bf r$$ For $F=(x^2y,3x^3z,yz^3)$, and ...
1
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4answers
99 views

I want to solve $\int \frac{2}{x^2(x^2+1)^2}dx$

I want to solve this primitive $$I=\int \frac{2}{x^2(x^2+1)^2}dx.$$ I substitute $u=x^2$ then, $$I=\int \frac{2}{x^2(x^2+1)^2}dx=\int \frac{du}{u^{3/2}(u+1)^2}=\cdots$$ How do I use partial fraction ...
0
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1answer
53 views

How do I evaluate $\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\lambda}} dt$ ??

How do you evaluate: $$\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\alpha}} dt = ??$$ Many thanks.
0
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1answer
36 views

Differentiating $- \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|x - n - y|^2}{4t^2}})dt$ wrt $x$?

I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect ...
1
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5answers
78 views

How can I solve $\int \frac{3x+2}{x^2+x+1}dx$

I want to compute this primitive $$I=\int \frac{3x+2}{x^2+x+1}dx.$$ I split this integral into two part: $$\int \frac{3x+2}{x^2+x+1}dx=\int \frac{2x+1}{x^2+x+1}dx+\int \frac{x+1}{x^2+x+1}dx,$$ For ...
1
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1answer
49 views

How can I calculate the integral $\int_M F^* \omega$?

I got stuck in the following problem. Let $M$ be the manifold defined by the equation $x^2+y^2+z^4=1$ and $F: M \to S^2$ defined as $F(x,y,z)=(x,y,z^2)$. I have to calculate the integral $\int_M F^* ...
3
votes
1answer
41 views

Modified Laplace's method

In the application of Laplace method (or steepest descent) it is often assumed that the dependence on the factor N, on which we are expanding the integral, is only in the argument of the exponential. ...
3
votes
1answer
40 views

How does Green's theorem apply here?

Let $D$ be the region delimited by $$\partial D: \begin{cases} C_1: x^2 + y^2 = 5^2\\ C_2:(x-2)^2+y^2= 1\\ C_3:(x+2)^2+y^2 = 1\\ C_4: x^2+(y-2)^2= 1\\ C_5: x^2+(y+2)^2= 1 \end{cases} $$ I've sketched ...
0
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0answers
14 views

Prove that a function is continuous (square integrability)

I need help for the following proof of continuity: Let $E=L_2([t_0,t_1],\mathbb R)$ be a Hilbert space of square-integrable real-valued functions on $[t_0,t_1]$. Let ...
2
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0answers
38 views

How to evaluate this integral with a bessel function

I have to evaluate the following two integrals that I would like to solve, Sin[θ] 2 π BesselJ[0, k Sin[θ] (μ[j] - μ[i])] Exp[2 k^2 σ^2 Cos[θ]^2], and ...
4
votes
4answers
158 views

Showing $\int_{1}^{0}\dfrac{\ln(1-x)}{x}dx=\dfrac{\pi ^{2}}{6}$

Is there way to show $\int_{1}^{0}\dfrac{\ln(1-x)}{x}dx=\dfrac{\pi ^{2}}{6}$ without using the Riemann zeta function?
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0answers
29 views

Find the expected value of the matrix

$\require{cancel}$ I want to see if I have solved this problem appropriately or not. If we have ...
0
votes
3answers
61 views

Evaluation of $\int_{0}^{\infty}t^3e^{-3t}dt$

I have to evaluate the integral $\int_{0}^{\infty}t^3e^{-3t}dt$ using complex analysis techniques (the laplace transform). Can you check my steps, please? $$\int_{0}^{\infty}t^3e^{-3t}dt =\Rightarrow ...
1
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1answer
51 views

How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$? To start ...
1
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1answer
36 views

How to use Substitution in an Abstract sense?

Based on my previous question: (somewhat related to it) $$\int f''(x^2)~dx$$ How would you go about and find the integral in an abstract sense as you can do the following with derivatives using ...
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votes
0answers
19 views

Difference between definity integral with constants and with variables

What is general difference between: $$\int_a^b f(x) \;\mathrm{dx}$$ $$\int_a^{x^2} f(x) \;\mathrm{dx}$$
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2answers
40 views

Integration by Parts? - Variable Manipulation

$$\int x^3f''(x^2)\,\mathrm{d}x$$ Solve using Integration by Parts. \begin{align} u&=x^3\qquad\mathrm{d}v=f''(x^2) \\ \mathrm{d}u&=3x^2\qquad v=f'(x^2) \\ &=x^3f'(x)-\int f'(x^2)3x^2 ...
0
votes
0answers
21 views

Showing a function has a global primitive on the unit disk minus the origin

I've reached a dead end for a problem with proving there is a global primitive for a continuous function on the unit disk D minus the origin with the condition that $\lim_{z\rightarrow 0}{zf(z)}=0$. ...
0
votes
1answer
51 views

How to find the following limit $\lim_{a \to \infty} \int \frac{e^{-(x-a)^2}}{\frac{1}{a^2}e^{-\frac{(x-a)^2}{2}}+e^{-\frac{x^2}{2}}} dx$

Does the following limit exists \begin{align} \lim_{a \to \infty} \int \frac{e^{-(x-a)^2}}{\frac{1}{a^2}e^{-\frac{(x-a)^2}{2}}+e^{-\frac{x^2}{2}}} dx \end{align} Observe that if we do \begin{align} ...
2
votes
1answer
32 views

Solve the improper integral: $\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$

I'm completely stuck on this one. I only know that it converges thanks to Wolfram, but I don't know how to evaluate it. $$\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$$ Thank you for the help.
3
votes
3answers
129 views

Why doesn't $\int_{-1}^{1}\frac{dx}{x} = \ln|x|\biggr\rvert_{-1}^{1} = 0$?

$1/x$ is an odd function, so it makes sense to me intuitively that the area would be $0$, and similarly I would expect that $\int_{-1}^{2}\frac{dx}{x} = \ln(2)$. Proof Wiki seems to confirm my ...
0
votes
1answer
23 views

Evaluatig a triple integral ( in Sphericals)

Hi I dont have much experiance with spherical coordinates, but to me it seems as if the following problem requires it. The problem I am having is finding the limits of integration. I want to solve ...
2
votes
0answers
18 views

integral derivation?

i have a problem in the partial derivatives of this function : so consider this integral : $u(x,t)=\int_0^t(\int_{\mathbb{R}^n}\phi_n(y,s)f(x-y,t-s)dy)ds$ can somone tell me why we have this ...
2
votes
2answers
67 views

Compute integral: $\int_0^{+\infty}\int_{-\infty}^{-x}\frac{1}{2\pi}e^{-\frac{1}{2} (x^2+y^2)}dx dy $

I would like to resolve this exercise: Let $W$ be a Brownian motion with $T_1=1 \text{ year}$ and $T_2=2 \text{ years}$. I want to compute the probability that $W_{T_1}$ be positive and $W_{T_2}$ ...
0
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2answers
46 views

Determine the general solutions of $x'''-5x''+8x'-4x=0 , t∈R$

I have found the following exercise in one of my courses: determine the general solutions of $x'''-5x''+8x'-4x=0 , t∈R$ How do you solve this?
0
votes
0answers
14 views

integral of trace function

How can I compute the below integral? $$ \int e^{-1/2*tr[(\mu\mu^T-\mu m^T-m\mu^T)\Sigma]}d\mu $$ in which $\mu \in R^{n},m \in R^{n},\Sigma \in R^{n*n}$ and $tr(.)$ is trace of matrix. I have ...
1
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1answer
39 views

Differential Equation: Can I further simplify it

This is a homework problem from course Differential Equation, which requires me to have Calculus 2 before taking it. $\frac{dy}{dt} = \cfrac{1}{2y+1}$ Here's my solution: $$ \int (2y+1) \; ...
1
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2answers
35 views

Problem integrating (substitution)

can you help me identify the mistake I'm making while integrating? Question: $$\int{\frac{2dx}{x\sqrt{4x^2-1}}}, x>\frac{1}{2}$$ my solution ...
-1
votes
0answers
48 views

Solving a system of nonlinear second-order differential equations with initial/boundary conditions.

I have developed a set of $n$ equations, $n$ variables for my dynamic system. The derivatives are second and first order in terms of $\theta$ (angle) of different components of the system (basically a ...
2
votes
3answers
54 views

How do you integrate $\csc^3(x)$? [duplicate]

How do you integrate $ \int \csc^{3}x\,dx $ ? I know that you have to change the integral into $\int \csc x (1 + \cot^2 x)\,dx$ But what do you do next?
1
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1answer
64 views

Compute $\int_0^{\infty} Q_1(y,b) \frac{y}{\sigma^2} \exp{(-y^2/(2\sigma^2))} \, dy$

We know that the first order Marcum Q-function can be represented as $$Q_1(y, b)=\int_{b}^{\infty} x \exp{(-(x^2+y^2)/2)} I_0(y x) \, dx ,$$ where $I_0(\cdot)$ is the modified Bessel function of the ...
1
vote
1answer
33 views

Why is $\frac{d(c*t)}{dt} = c + c'*t$?

I have found in one of my courses the following equality however I don't understand why is it equal. $$\frac{d(c*t)}{dt} = c + c'*t$$