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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1
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1answer
34 views

Low storage 3rd order Runge-Kutta scheme

I'm looking for a 3rd order Runge-Kutta scheme of order 3 (or higher) with low-storage, which means that not all intermediate results must be stored concurrently. I found an old paper which presents ...
1
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1answer
48 views

$\int_0^1\frac{1}{r}\frac{1}{\left[\log\left(1+\frac{1}{r}\right)\right]^n}dr$ finite?

Does someone have a hint for me why the integral $\int_0^1\frac{1}{r}\frac{1}{\left[\log\left(1+\frac{1}{r}\right)\right]^n}dr$ is finite? $n$ is a natural number greater than $1$.
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2answers
63 views

Showing that $\frac{d\mu}{d\nu} = 1/\frac{d\nu}{d\mu}$

I am trying to prove the following statement: Suppose $\mu$ and $\nu$ are finite measures on the measurable space $(X,\mathcal A)$ which have the same null sets. Show that there exists a ...
0
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1answer
31 views

Explanation for the proof of the fundamental theorem of calculus

Let $F(x) = \int_a^x f(x)\ dx$. The proof starts like this: $$\frac{F(x_0+h) - F(x_0)}{h} = \frac{1}{h}\cdot \int_{x_0}^{x_0+h} f(x)\ dx = f(x_0) + \int_{x_0}^{x_0+h} \frac{f(t)-f(x_0)}{h}\ dt$$ Why ...
0
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2answers
65 views

Integration of $e^{-x}$ with respect to y

I'm not sure if I'm being incredibly stupid and having a brain dead moment! any help is appreciated! The question I'm referring to is dealing with the integration of an exponential function of x with ...
1
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0answers
101 views

Finding “Solution” of the partial diffrential equation $z=px+qy+f(p,q)$

Adding more to the title: $p=z_x,q=z_y$ My attempt:On applying charpits method,I get $dp/0 = dq/0 \implies $ $p=a,q=b$ so that $$dz=a \, dx +b \, dy \implies z=ax+by+c$$ Putting for $z$ in the ...
1
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4answers
90 views

What is the difference between a Summation and an Integration?

What is the difference between a Summation and an Integration? Both of them add some values. Right? Then what is the difference? Please, explain in layman's terms.
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1answer
29 views

$f(x) = \frac{1}{x^a+x^b}$ or $f(x) = \frac{1}{(x^a+x^b)^p}$?

Let $0< a\leq b$, for which values $p$, does the function: $$f(x) = \frac{1}{x^a+x^b}dx$$ belong to $L_p(0,\infty)$? The first step in the given (potentially incorrectly written) solution is: ...
1
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1answer
40 views

Exchange order of partial differentiation and integration.

Consider a two-valued function $f(x,y) : R^2 \rightarrow R$. Define $f_x(x,y) = \frac{\partial f(x,y)}{\partial x}=\lim_{\epsilon\rightarrow 0}\frac{f(x+\epsilon,y)-f(x,y)}{\epsilon}$ and $f_y(x,y)$ ...
1
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3answers
80 views

Integrate by using trigonometric substitution. [closed]

Hello I was wondering if I could have some assistance on integrating the following integral using trigonometric substitution. $$\int\Big(\sqrt{4x^2+9-4x}\Big)^3\,dx.$$
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1answer
85 views

If $a_{mn}\ge 0,\,$ then $\,\sum_{m\in\mathbb N} \sum_{n\in\mathbb N} a_{mn} = \sum_{n\in\mathbb N} \sum_{m\in\mathbb N} a_{mn}$. [closed]

I am studying Measure Theory, and in particular, integration of non-negative measurable functions. I have encountered the following problem: If $\,a_{mn}\ge 0,\,$ for all $\,m,n\in\mathbb N,\,$ then ...
4
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1answer
59 views

Using Cauchy's integral theorem to prove a inequality

I am trying to solve this question: Let $f(z) = c_0 + c_1z + \ldots + c_nz^n$ be a polynomial. If the $c_k$'s are real, show that $$\int_{-1}^1 f(x)^2 dx \le \pi \int_0^{2\pi} \left | ...
1
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1answer
35 views

What did they do to get -2/3 as the constant for this integral?

Find the integral of: $$\int \frac{1}{\sqrt{x}(1-3\sqrt{x})}dx$$ So far I know to substitute $u = (1-3\sqrt{x})$ and $du = \frac{-3}{2\sqrt{x}}dx$ but I'm not sure what to do with the $\sqrt{x}$ in ...
0
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1answer
19 views

Find a common denominator

I'm trying to integrate a function but first I need to find a common denominator for: $\frac{A}{x-8}\ $ + $\frac{B}{x+1}\ $ + $\frac{C}{x-1}\ $
2
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1answer
13 views

Inequality on an integrable function

Suppose $f:X \to \mathbb{R}$ is an integrable function and $(X,F,m)$ a measure space. For all $C\in F$ with $m(C)>0$ $$\left| \frac{1}{m(C)}\int_C f dm\right|\le L$$ where $L>0$. Need to show ...
1
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1answer
60 views

Integrating $\int_{1}^{\sqrt{3}} \sqrt{1+\frac{1}{t}}dt$ [closed]

I just started my calculus 3 class 2 years after taking my calculus 2 class and I'm having trouble remembering how to solve some integrals. This one below is really confusing me as I couldn't find ...
1
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1answer
46 views

Using trig to integrate $x^2/\sqrt{16-x^2}$

I'm trying to integrate: $\int\frac{x^2}{\sqrt{16-x^2}}\ dx$ My try was to convert $x$ to $4\sin(u)$ and $dx$ to $4\cos(u)du$, but I'm not sure. Thanks.
1
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0answers
30 views

Integral equations iterates question

Suppose that $K(x,y) = g(x)h(y)$ and that $\int_{a}^{b} g(x)h(x) dx=0$. Let $\psi_0(x)=f(x)$. Show that all iterates equal the first iterate and find a simple formula for the solution. I'm really ...
4
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2answers
106 views

To prove the integral inequality $\int_\overline{\theta}^{\pi}\frac{d\theta}{\sqrt{1-\lambda \cos{\theta}}}\gt\pi$

The following inequality comes up in connection with motion in a dipole field. One has to show that $$\int_\overline{\theta}^{\pi}\frac{d\theta}{\sqrt{1-\lambda \cos{\theta}}}\gt\pi$$ where ...
0
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1answer
42 views

Proving that the solution to $yu_x-xu_y=0$ containing $x^2+y^2=a^2,u=y$ doesn't exist.

This is basically a Cauchy problem.Parametrizing the given curve(Initial curve):$x=a\cos s,y=a\sin s,u=a\sin s$ $y'(s)y(s)+x(s)x'(s)=a\cos s(a\sin s)-(a\cos s)(a\sin s)=0 \implies$ Characteristic ...
0
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1answer
26 views

Path Integral over circle always $0$

Let's say we want to evaluate the integral $$ \int_{\lvert x \rvert = R} f(x) dx $$ where $R \gt 0$ is the radius of a circle. Now we parameterize $\varphi(t) : [0, 2\pi] \rightarrow D $ : $$ ...
1
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2answers
68 views

Integrating $\sqrt{x^2+a^2}$

I'm trying to integrate this function wrt $x$, substituting $x = a \tan \theta$ $$ \int \sqrt{x^2+a^2} dx = a^2 \int \frac {d\theta}{\cos^3\theta} = $$ $$= a^2 \cdot \frac 12 \left( \tan\theta ...
1
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1answer
29 views

Trapezium rule: undefined values of $f(x)$

I am trying to estimate the area between the curve $f(x) = \frac {\sin(x)}{x^2+2x}$ and the $x$ axis between $x=-1$ and $x=2$ using the trapezium with $6$ strips. However, when calculating values for ...
2
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1answer
45 views

Calculate surface area of flat figure by using double integral and polar coordinates

Check me please. I tried check it via WolframAlpha, but I don't trust in it 100%. Task: Calculate surface area of flat figure by using double integral in polar coordinates. Figure confined by line: ...
1
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2answers
92 views

Evaluating the indefinite integral $\int \tan \sqrt {x} \,dx$

$$\int \tan \sqrt {x} \,dx$$ I was trying to solve this. But it took very long time and three pages. Could someone please tell me how to solve this quickly.
0
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3answers
138 views

Evaluating:$\sum_{n=0}^\infty\frac1{\binom{2n}{n}}$ [closed]

How to evaluate: $$\sum_{n=0}^\infty\frac1{\binom{2n}{n}}$$ $\binom{n}{r}$ is the binomial coefficient. If possible, present different methods as well.
1
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1answer
39 views

To test the convergence of $\int_0^{1} \frac{x^p \log x}{1+ x^2} dx$

To test the convergence of the improper integral: $$\int_0^{1} \frac{x^p \log x}{1+ x^2} dx$$ Here we see that $0$ is the point of infinite discontinuty for $p<0$. Let $f(x) = \frac{x^p \log ...
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votes
3answers
72 views

How to compute $\int {\frac{1}{{4 - 9{x^2}}}dx} $? [duplicate]

How can I evaluate the following integral $$\int {\frac{1}{{4 - 9{x^2}}}dx} $$
0
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1answer
28 views

Solving Differential Equation about rate of infected computers

I am having some trouble solving this differential equation for the rate of infected computers in a botnet at time t $$\frac{\mathrm{d}x }{\mathrm{d} t} = \frac{1}{c\nu (1-x) + \beta x(1-x) - \gamma ...
3
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2answers
30 views

Evaluate the inward flux of the vector field $F=<y,-x,z>$ over the surface $S$ of the solid bounded by $z=\sqrt{x^2+y^2}$ and $z=3$.

Evaluate the inward flux of the vector field $F=<y,-x,z>$ over the surface $S$ of the solid bounded by $z=\sqrt{x^2+y^2}$ and $z=3$. this is basically an inverted cone (right?) So by changing ...
2
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3answers
42 views

Surface area of $x^2+y^2+z^2=9$, where $1\leq x^2+y^2\leq4$ and $z\geq0$

Let $S$ be the portion of the sphere $x^2+y^2+z^2=9$, where $1\leq x^2+y^2\leq4$ and $z\geq0$. Calculate the surface area of $S$ Ok i'm really confused with this one. I know i have to apply the ...
7
votes
1answer
71 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
1
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1answer
27 views

Question about Integral with exponential function

Please refer to the image below. I would like to ask why the highlighted part would be gone in step $2$ ? What calculation involved making it $0$ ? Thank you so much!
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2answers
58 views

What value of $N$ to use in Simpson's rule to reach desired accuracy?

I need to calculate Simpsons rule for the integral of $$\frac{e^x-1}{\sin x}$$ from $0$ to $\pi/2$ with minimum number of intervals $N$ up to $10^{-6}$ accuracy. Wolfram alpha seems to be giving me a ...
0
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1answer
42 views

What is my mistake: Asymptotic behaviour of the following integral?

Okay, I am going to be very specific. I have the following integral $$\int_{-1}^1 \mathop{dx}\frac{x^{n-2m}(a^2+x^2)^{(k-2)n/2+(3-k)m/2}}{(c_1 ...
1
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1answer
28 views

Definition and existence of Riemann integral in PMA Rudin

I understand all moments bseides $(1)$ and $(2)$. Why Rudin considers $\inf U(P,f)$ and $\sup L(P,f)$. Why we can't considered $\sup U(P,f)$ and $\inf L(P,f)$? Can anyone explain it to me please?
2
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1answer
69 views

Proving $\int_0^{\infty} \frac {x^{m-1} - x^{n-1}}{1-x} dx$ and $\int_0^{\infty} x^m (\log x)^m dx$

Prove that: (a) $\int_0^{\infty} \frac {x^{m-1} - x^{n-1}}{1-x} dx$ is convergent if $0<m<1$ and $0<n<1$; (b) $\int_0^{\infty} x^m (\log x)^n dx$ is convergent if $m< -1$; Getting no ...
1
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2answers
55 views

Exact value of integral $\int_{0}^{\pi/4}(\sec x-x)(\sec x+x)dx$

In terms of integration, how would you obtain the "exact-value" of $$\int_{0}^\frac\pi4(\sec x-x)(\sec x+x)dx.$$ Note: $1+\tan^2x=\sec^2x$
1
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1answer
19 views

estimates on an improper integral associated with normal distributuion

Show that $\int_{x}^{\infty}e^{-\frac{t^2}{2}}dt\geq e^{-\frac{t^2}{2}}(\frac{1}{x}-\frac{1}{x^3}) $ for all positive $x$ Does it require the mean value theorem, or the Taylor series expansion? It is ...
1
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2answers
36 views

Calculate double integral using Polar coordinate system

Need to calculate $\int_{0}^{R}dx\int_{-\sqrt{{R}^{2}-{x}^{2}}}^{\sqrt{{R}^{2}-{x}^{2}}}cos({x}^{2}+{y}^{2})dy$ My steps: Domain of integration is the circle with center (0,0) and radius R; $x = ...
0
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0answers
17 views

Integral alternative to $\text{Ei}(x)$ function.

Is there an alternative but simple way to find the integral of the following function. I tried to find by the method of "Integration by parts" but it again and again result the answer same as ...
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1answer
17 views

Evaluating a contour-integral.

Consider the ellipse $C$ given by $x^2 + y^2/4 = 1$. How to evaluate $$\int_C x^2 \, \nu(d(x,y))$$ where $\nu$ is the Lebesgue length measure on $C$? I am not sure if this can be computed like a ...
2
votes
3answers
140 views

Integrate $\sin(\cos(x))$ with respect to $x$ [duplicate]

Solve this: $$\int\sin(\cos x)dx$$ I checked on Maxima, mathematica but both cannot find its integral though numerical approximation is available in later. Has someone faced similar problem? ...
1
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1answer
48 views

How to show continuously differentiable.

Let $$f(x)=\int_{0}^{\infty} e^{-xt} t^x\ \mathsf dt$$ for $x>0$. Show that $f$ is well defined and continuously differentiable on $(0, \infty)$ and compute its derivatives. My confusion is ...
0
votes
1answer
52 views

Integral inequalities.

If f(t) is non-negative and bounded function, then by Schwartz inequality, we have $(\int\limits_0^tf(s)ds)^2 \leq t\int\limits_0^tf^2(s)ds$. Now my questions are, (i) is there any possibility to ...
0
votes
0answers
8 views

using Green's Theorem to calculate the Work done for a vector function.

$$f_a(x,y) = \frac {1}{(x^2+y^2)^a}(-y,x)$$ is a vector. Q is a square $$[-1,1]\times [-1,1]$$ and R also a square $$[1,2]\times [-1,1]$$ How do i calculate the Work Integral about Q and R? of the ...
0
votes
1answer
29 views

About the convergence of the improper integrals:

About the convergence of the improper integrals: 1: $\int_0^{\pi/2} \frac{1}{e^x - \cos x} dx$ 2: $\int_0^{\pi} \frac{1}{\cos \alpha - \cos x} dx, 0 \leq \alpha \leq \pi.$ In the first problem $0$ ...
2
votes
1answer
28 views

Indefinite integral problem: $\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ [duplicate]

$ I =\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ my attempt: $I=\int_1^n \frac{(x - [x])^{[x]}{[x]}}\implies\sum_{i=1}^n\int_r^{r+1} \frac{(x-r)^r}{r}dx $ Now by ...
2
votes
1answer
59 views

Evaluating $\int_0^u\int_0^{2u-x_1}\cdots \int_0^{ku-x_1-\cdots -x_{n-1}}dx_ndx_{n-1}\cdots dx_2dx_1$

I have reduced a problem to evaluating the integral $\int_0^u\int_0^{2u-x_1}\cdots \int_0^{ku-x_1-x_2-...-x_{n-1}}dx_ndx_{n-1}\cdots dx_2dx_1$. I tried computing this for small $n=1,2,3$ but I can't ...
3
votes
3answers
88 views

Evaluate $\int \frac{dx}{e^{2x}+1}$

Evaluate $$\int \frac{dx}{e^{2x}+1}$$ Here's what I did: $$t=e^{2x} \Rightarrow \frac{1}{2}\int \frac{dt}{t(t+1)} = \frac{1}{2}\left(\int \frac{dt}{t} - \int \frac{dt}{t+1}\right) = ...