Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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2
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0answers
18 views

Computing the limit $\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$ for fixed $n \in \mathbb{N}$

I'm working on a problem that asks to compute $$\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$$ for fixed $n \in \mathbb{N}$. What I've tried so far is to do a $u$-...
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0answers
17 views

factoring out total derivative from integral

What is the easiest way to factor out the total derivative from the following integral? I show my attempt and method below. $$ I=\int_a^b dt \int_a^b dx \, \frac{df(t)}{dt} \left[g(x,t)h(x,t)+g^2(x,t)...
0
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1answer
43 views

Any software recommendations to draw such graphs?

I am looking any alternatives software to draw similar graph images i posted here.
2
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1answer
38 views

A curious derivation of the Gaussian integral from Plancherel's Theorem

I was playing around with the Plancherel's theorem and stumbled across a derivation of the Gaussian integral which I have never seen before. Could folks check to see if derivation is correct? If so, ...
0
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0answers
29 views

Mean value theorem on the open interval

Mean value theorem on an open interval Function $g(x)$ is non-negative and $g(x)\in R[a;b]$, function $f(x)$ is continuous on the an open interval $(a;b)$ and a product $f(x)g(x)$ is Riemann-...
-1
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1answer
53 views

Proving $\int_0^{2\pi}\frac{\sin x}{x}\,dx>0$ and $\int_0^\sqrt{2\pi}\sin x^2\,dx>0$

Prove that $$\int_0^{2\pi} \frac{\sin x}{x} \, \mathrm{d}x \gt 0 \tag1$$ $$\int_0^\sqrt{2\pi} \sin x^{2} \, \mathrm{d}x \gt 0 \tag2$$ For the first integral I used the Mean value theorem but I didn't ...
0
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0answers
22 views

Integration and kth derivative

Let $f_{0}$ be continuous on $[0, b]$. For $k$ in $\mathbb{N}$ define $$ f_{k}(x)=\frac{1}{k !} \int_{0}^{x}(x-t)^{k} f_{0}(t) d t. $$ Show that for every $k$, the $k$th derivative of $f_{k}$ exists ...
1
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1answer
23 views

Integration and Limit for sine

Fix any $a$ in $(0, \pi)$. For each $k$ in $\mathbb{N}$ define the sequence $s_{k}=\int_{a}^{\pi}(\sin k x) / k x d x$. Prove that $\lim _{k \rightarrow \infty} s_{k}=0$ In order to tackle such ...
-1
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1answer
15 views

Multiplication and sum of Riemann integrable functions

Consider, that $g(x)$ and $f(x)$ two Riemann integrable functions. There is $s(x) = g(x)*f(x)$. Will $s(x)$ be also Riemann integrable? The same question about sum $t(x) = g(x)+f(x)$. It seems to me, ...
0
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3answers
45 views

Finding graph corresponding to $\int_0^{\sqrt{x} } e^{ -\frac{u^2}{x} } du$

Question 36: Finding graph corresponding to $\int_0^{\sqrt{x} } e^{ -\frac{u^2}{x} } du$ $x>0$ and $f(0)=0$ Clearly we can't say the function is increasing or decreasing just by inspection ...
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1answer
22 views

Setting the limits of integration in the triple integral for flux

Let $T$ be the figure bounded by: $y-\sqrt{1-x^2-z^2}=0$, $x^2+z^2=1$, $y-z=2$(green axis is y, blue - z) I need calculate the lateral flow for semi-sphere(that burgundy thing on the image) $\vec a = ....
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0answers
31 views

Integral of a function $A$ on a manifold… how does $A$ transform under coordinate changes?

I understand how to do a change of coordinate on a volume form: $$ \begin{align} V&=\int \sqrt{|g|} dx \wedge dy\\ &=\int \sqrt{|g|}|\det F| du \wedge dv\\ &=\int \sqrt{|g'|}du \wedge dv \...
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0answers
25 views

Oriented boundary integral over $G\colon |z-i| < 2$

Let's assume we have a domain $G\colon |z-i| < 2$ Now we want to calculate the following oriented boundary integral: $$\int_G \left[\frac{e^z}{z^2 (z^2+4)} + \frac{3}{4} z^2\sin{\frac{2}{z}}\right]\...
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0answers
24 views

Find the volume between the two surfaces. [duplicate]

Find the Volume of: $x^2+y^2\le z\le \sqrt{6-x^2-y^2}$ My Attempt: So I need to find $\iint_D(\int^{\sqrt{6-x^2-y^2}}_{x^2+y^2}dz)dxdy$, while $D$ is the area of the intersection between the two ...
1
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1answer
21 views

Integral when absolute inequality holds

Suppose that $f_1,f_2:[0,\infty) \rightarrow ℝ, f_1,f_2 \in \mathcal{R}$, $\lvert f_1(x) \rvert \leq f_2(x) $ for all $x \geq 0 $. Prove that $\int_0^\infty {f_1(x)\mathrm dx}$ converges if $\int_0^\...
2
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0answers
34 views

Compute the following double integral. [closed]

Supposing existence I have to calculate: $$\int_{0}^{1}{\int_{1}^{\mathrm{e}^{1+x}}{\frac{\mathrm{e}^x}{\sqrt{x \ln y}}\,\mathrm{d}y}\,\mathrm{d}x}$$ I tried a change of variables but I didn't find ...
2
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2answers
101 views

Does $\int_a^bf(x)\,\mathrm dx$ always exist and equal $F(b)-F(a)$?

Consider, you have $F'(x) = f(x)$ for all $x$ from $[a,b]$, where $F(x)$ is antiderivative. Does $\int_a^bf(x)\,\mathrm dx$ necessarily exist and is equal to $F(b)-F(a)$? From condition we could ...
1
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1answer
16 views

Show the following piecewise function is integrable

Let$$f: [0,7] \to \mathbb{R}, f(x) = \begin{cases} 5 , 0 \le x < 4 \\ 0, x = 4 \\ 4, 4 < x \le 7\end{cases}$$ Consider the partition $\lbrace 0, 4 - \frac{1}{n}, 4 + \frac{1}{n}, 7\rbrace$ Show ...
1
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1answer
22 views

improper integration of a cosine function

Prove that $\int_0^\infty{\cos{(t^\alpha)}dt}$ converges when $\alpha>1$. I tried to use the Taylor theorem to expand the cosine as $\cos{(t^\alpha)}=\sum_{n=1}^\infty{\frac{(-1)^{n}{t}^{2\alpha n}...
1
vote
3answers
72 views

What is the derivative of $\int_{x^3}^{x^2} e^{y^2} dy$? [closed]

My only idea is to calculate this integral but I know it is very hard to calculate it so there must be some smarter way. I will appreciate any hint or help.
2
votes
3answers
87 views

Integration for exponential

Prove that $$\lim _{x \rightarrow \infty} e^{-x^{2}}\int_{0}^{x} e^{t^{2}} d t=0$$ What is the main intuition to tackle this problem? I tried to utilize all the possible theorems, including Taylor's ...
1
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1answer
36 views

Exponential Function and Integral

For any fixed $x>0$, find the value $$ \lim _{k \rightarrow \infty} \int_{0}^{x} \exp \left(-k t^{2} / 2\right) dt. $$ From the first sight, it seems that the function $f_k(t) = \exp(-k t^{2} / 2)$ ...
0
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0answers
21 views

Question on using Fubini iteratively

Let $f(x_1, y_1, x_2, y_2)$ be a measurable function, $\mu$ and $\lambda$ two $\sigma$-finite measures on spaces $X$ and $Y$ respectively. Then if it holds that $$ \int_X \int_Y |f(x_1, y_1, x_2, y_2)|...
3
votes
2answers
110 views

My solution to $\int \frac{2x^2}{x^2+4}dx$

$$\int \frac{2x^2}{x^2+4}dx$$ Attempt. I have actually done this integral but I'm wondering if what I did was correct as I have a different answer from WolframAlpha. Here it is: $$\int \frac{2x^2}{x^2+...
3
votes
1answer
67 views

“Find the volume of the ellipsoid $x^2+\frac{y^2}{100}+\frac{z^2}{4} = 1$” was the question and here's my attempt.

So, I asked this question and got downvotes and someone saying I should present my attempt. Here lies my thought process: I know the volume is $\iiint dxdydz$ on any W$\subset R^3$ and I "just&...
1
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1answer
76 views

$\int_0^\infty \frac{x^a}{(x^2+1)^2}dx$

The following exercise is from the book of Churchill of complex analysis. Solve the integral $$\int_0^\infty \frac{x^a}{(x^2+1)^2}dx$$ where $-1<a<3$ the book of complex analysis of Churchill ...
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1answer
44 views

Find the volume of the ellipsoid $x^2+\frac{y^2}{100}+\frac{z^2}{4} = 1$ [closed]

If you have any ideas, please let me know.
3
votes
2answers
49 views

proving an integral inequality_

I'm having problem proving the following integral inequality. Let $f$ be a continuous function on $[0,1]$ such that $f(0)=0$ and $f'(x)\geq1$ for all $x\in(0,1)$. Show the following inequailties hold. ...
1
vote
1answer
20 views

Integration with affine transformation

Context: I am trying to understand whether the $L^{2}$ function space with inner product $$\langle g_{1}, g_{2}\rangle=\int\limits_{\mathbb{R}^{n}}g_{1}(x)g_{2}(x)e^{-x^{\top}x}\,\mathrm dx$$ is ...
1
vote
1answer
70 views

What's wrong with the proof that $1!=0$

I found this proof which shows that $1! = 0$ using Taylor Series. Here's what we do: Start with the Taylor series of $\sin(x)$. We get : $$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}...
2
votes
1answer
51 views

Find an estimate for $\int_{\pi/2}^\pi \sin(x) dx$ using the Monte Carlo Simulation

I want to find an estimate for $$\int_{\pi/2}^\pi \sin(x) dx$$ I want to use the monte carlo simulation method. I've plotted the graph of $\sin(x)$ in the given interval. The total area $$\begin{align*...
0
votes
0answers
13 views

Underestimation of an indicator function

I am asked to determine the underestimation of an indicator function on the interval [0,2], defined as follows: $\int_{0}^{2} \mathbf{1}_{1}\,dx$. Intuitively, I'd say the answer is 0, since we have ...
3
votes
1answer
32 views

Solution Verification: Find the mass of $V$. (Triple integral)

We define $V=\{(x,y,z)\in\mathbb{R}^3: x^2+y^2+z^2\le z$ } Mass Density is given by $g(x,y,z)=\sqrt{x^2+y^2+z^2}$ , Find the Mass of $V$. My Work: So I need to find $I=\iiint_{V}\sqrt{x^2+y^2+z^2}...
0
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2answers
45 views

Integral on the intersection of a cone and cylinder

Let $S$ be the surface formed by intersection of the cone $ z = \sqrt{x^2 + y^2}$ and the cylinder $x^2 + y^2 = 2x$. Calculate the following integral over the surface $S$. $$\iint_S x^2y^2+ y^2z^2+ x^...
0
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1answer
23 views

Figuring out the integral of a function that is defined differently in a single point than the rest of the domaine

This is my first post and English is not primary language, so I'm sorry if I mess it up. So my question is if i have a function that is, for example, defined like this: $$ f(x) = \begin{cases} x-4 &...
1
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0answers
60 views

Interesting find in integrating $f(\text{x})=e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right).$ Why?

Interesting find in integrating $f(\text{x})=e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right).$ Why? It looks like for $$f(\text{x$\_$})=e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right),$$ $$-\frac{...
1
vote
1answer
48 views

Integration of $3x^2ydx+x^3dy$ by two methods

Find $\int(3x^2ydx+x^3dy)$ $\int(3x^2ydx+x^3dy)=\int d(x^3y)=x^3y+c$ Or, $\int(3x^2ydx+x^3dy)=\int3x^2ydx+\int x^3dy=x^3y+x^3y+c=2x^3y+c$ Why am I getting different answers from the two methods? I am ...
0
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0answers
10 views

Positive definite Hessian matrix

1)If I want to prove a Hessian matrix is positive definite and it contains the entries of $x=(x_1,...,x_n)$, can I multiply it also with the same $x$ to simplify my calculations? 2)Is the integral of ...
0
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1answer
59 views

Verify the inequality without evaluating the integrals [closed]

Use the properties of integrals to verify the inequality without evaluating the integrals. Explain why. (Use the graph. $f(x)=x^2+1$) $$\cfrac{3}{4}\le\int_0^1{e^{x^2}}\,\mathrm dx\le{e}$$ I just ...
3
votes
1answer
47 views

Infinite sum of iterated integrals of matrix products

The problem: Let $$N(z) = \begin{pmatrix} 0 & \frac{1}{z} \\ \frac{1}{2(z-1)^2} & 0 \end{pmatrix} \;.$$ I write this post to seek any hints about how to compute the following infinite sum over ...
10
votes
2answers
180 views

Find $\int _0^1\frac{12\arctan ^2 x\ln (\frac{(1-x)^2}{1+x^2})-\ln ^3(\frac{(1-x)^2}{1+x^2})}{x}\:dx$

I want to find and prove that: $$\int _0^1\frac{12\arctan ^2\left(x\right)\ln \left(\frac{\left(1-x\right)^2}{1+x^2}\right)-\ln ^3\left(\frac{\left(1-x\right)^2}{1+x^2}\right)}{x}\:dx=\frac{9 \pi ^4}{...
2
votes
1answer
50 views

$ \int^{\frac{\pi}{2}}_{0} \cos^nx \ dx = \frac{n-1}{n} \int^{\frac{\pi}{2}}_{0} \cos^{n-2}x \ dx$ for $n \in \{ \mathbb{N} > 2 \}$ proof [duplicate]

I need to prove that: $$ \int^{\frac{\pi}{2}}_{0} \cos^n x \, \mathrm dx = \frac{n-1}{n} \int^{\frac{\pi}{2}}_{0} \cos^{n-2} x \, \mathrm dx$$ for $n \in \{\mathbb{N} > 2\}$ I know that the ...
2
votes
1answer
48 views

Problem 11, Chapter 18 of Calculus by Spivak

The problem: Let $f$ be a nondecreasing function on $[1;\infty)$. Define $F(x)$ as follows: $$F(x) = \int_{1}^{x}\frac{f(t)}{t}dt.$$ Prove that $f$ is bounded on $[1;\infty)$ if only if $\frac{F}{\log}...
0
votes
0answers
5 views

How to calculate a volume integral from a data file containing $x$, $y$ and $z$ coordinates and electric field value at each $(x,y,z)$?

I have a data file from a simulation that contains $E$-field values at different coordinates $(x,y,z)$. I want to calculate the integral $\int E^{2} dV$ over a cylindrical volume in Python or MATLAB. ...
2
votes
1answer
51 views

Evaluate integral in terms of the hypergeometric function

Mathematica's Integrate() gives the following result: $$ \int \frac{1}{x^{b}}\left[1-(d/x)^b\right]^{N-1}\,\mathrm dx= -\frac{x^{1-b}{_2F_1}\left(\frac{b-1}{b}, 1-...
0
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0answers
15 views

Different notation in Expectation's integration in (Empirical) Risk Minimization

For the philosophy of Statistical Learning Theory, the Risk Minimization is the focus. We want to minimize $$ R(f)=\mathbb{E}[L(f(x),y)]=\int~L(f(x),y) ~dP(x,y), $$ where $P(x,y)$ is the "joint ...
1
vote
1answer
33 views

$\lim_{n \to \infty} \sum_{k=0}^n \frac{\sqrt{2n^2+kn-k^2}}{n^2}$ the problem of $k = 0$. Am I doing it right?

I would like to calculate: $$\lim_{n \to \infty} \sum_{k=0}^n \frac{\sqrt{2n^2+kn-k^2}}{n^2}$$ We have that: $$\lim_{n \to \infty} \sum_{k=0}^n \frac{\sqrt{2n^2+kn-k^2}}{n^2} = \lim_{n \to \infty} \...
1
vote
2answers
76 views

Antiderivative of $e^{-|x|}$

Splitting the function into the negative and positive parts of the domain of $x$ gives $$ e^{-|x|}= \left\{ \begin{aligned} &e^x, &x ≤ 0 \\ &e^{-x}, &x > 0, \end{aligned} \right. ...
1
vote
1answer
30 views

Given a general region evaluate the double integral over that region.

Consider the integral $$ J = \iint_T \log(1+x^2)\,dx \,dy, $$ where $T$ is the region that is bounded $y=0$, $x=y$ and $x=2$. My attempt: $$\int^{x=2}_{x=0} \, \int^{y=x}_{y=0} \log(1+x^2) \,dy\,dx $$ ...
1
vote
1answer
35 views

Prove if we have for all $0 \leq k \leq n, \, \int_0^1 x^k f(x) \, \mathrm{d}x =0$ then $f$ has at least $n+1$ zeroes.

Let $n\in \mathbb{N}$ and $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that, $$\forall 0 \leq k \leq n, \,\int_0^1 x^kf(x) \, \mathrm{d}x=0$$ prove that $f$ has at least $n+1$ zeroes. What I've done ...

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