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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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35 views

Fundamental Theorem of Calculus improper integral question

I want to find the derivative of a function G(x) defined by: $$ G(t) = \int_{0}^{t^2} x^{2} \sin(x^{2}) \, dx$$ Am I correct in evaluating the answer to be $= 2t^{3}\sin(t^{2})$? What I did was ...
1
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1answer
18 views

Finding the function when given definite integral output

Find a function $f$ and a positive number $a$ such that: $$\int_{\sqrt y}^a f(t)\ln(t)\, dt= {\exp(y)\over2}-\ln\left({\sqrt y\over a}\right)-\pi$$ for all $y>0.$
3
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2answers
27 views

Evaluating $\int_{} \frac{xe^{2x}}{(1+2x)^2}dx$ via integration by parts

$\int_{} \frac{xe^{2x}}{(1+2x)^2}dx$ I am having trouble picking the correct $u/dv$ before integrating by parts. I felt like L.I.A.T.E. did not really help me here... This is what I tried, but it ...
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1answer
42 views

Integrating an interesting exponential!

I'm trying to integrate $$\int_{x}^{\infty} e^{-t^\beta} dt$$ where $\beta \in (0,1)$. The form looks similar to that of an incomplete gamma function, how do I proceed? I try substitute $t=u^\...
1
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1answer
36 views

Limit of some integrals.

Is this true: $$\frac{\int_{-\delta }^{\delta }(1-x^2)^ndx}{\int_{-1}^{1}(1-x^2)^ndx}\rightarrow 1$$ for a previously fixed $\delta\in(0,1)$?
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0answers
19 views

Definition of Dirichlet energy

I don't understand definition of Dirichlet energy $$ E[u]:=\frac{1}{2} \int_\Omega \Vert \nabla u \Vert^2 $$ for $u:\Omega \to \mathbb R^n$ with $\Omega \subset \mathbb R^n$. Let's consider for ...
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0answers
12 views

How to find the Current in terms of C for the equation in description

For $${V=C\cdot sin(120\cdot \pi\cdot t)}$$ where $$t$$ is time (in seconds) and is a constant. The square root of the average value of $$V^2$$ over one period (or cycle) of $$V$$ is called the root-...
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0answers
45 views

Indefinite integral of $\int\cos(x)\cos(\frac a x)dx$

I've tried integration by parts. I can integrate both factors: $\int\cos(x)dx = \sin(x) + C_1$ $\int\cos(\frac a x) = a Si(\frac a x) + x \cos(\frac a x) + C_2$ However I'm stuck at this point. I ...
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3answers
54 views

Can you spot the mistake?

Let $$I=\displaystyle\int\sin^2x \ dx$$then $$I=\displaystyle\int(1-\cos^2x) \ dx=x+C-\displaystyle\int \cos^2x \ dx$$ Using the substitution $x=x+\frac{\pi}{2}$ we get $$I=x+C-\displaystyle\int\...
3
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3answers
95 views

How can I prove the following integral equality?

Let $f:[a,b] \to \Bbb R $ be a continuous function. Prove that there's a c, $c \in (a,b)$ with the following property: $$\int_a^btf(t)dt=a\int_a^cf(t)dt+b\int_c^bf(t)dt$$ Thanks in advance.
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0answers
43 views

Stuck on an integral while solving an ODE

I was solving $$\left(1+3e^{y/x}\right)\,\mathrm dx+3e^{y/x}\left(1-\dfrac xy\right)\,\mathrm dy=0\\ \implies \dfrac{\mathrm dy}{\mathrm dx}= \dfrac{1+3e^{y/x}}{3e^{y/x}\left(\dfrac xy-1\right)}$$ ...
2
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0answers
12 views

Using the Fundamental Calculus Theorem for two variables to prove smoothness.

There is a probability density function that depends on non-deterministic ($v$) and random ($x$) parameters: $Pr(v)=\int_{G(v)} dP(v)$, where $G (v)$ is the "goal" region, the probability of getting ...
2
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2answers
48 views

An Integral Error

I was studying the derivations for the volume and surface area of a sphere . One derivation , for the volume of the sphere is the disk-method . A circle of radius $r$ is considered , centred at ...
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2answers
27 views

Prove that there is a compact $K_{\epsilon}$, $J$-measurable s.t. $\int_{A}\chi_{A-K_{\epsilon}} < \epsilon$

Let $A$ be a $J$-measurable set and $\epsilon > 0$. Prove that there is a compact $K_{\epsilon} \subset A$, $J$-measurable such that $$\int_{A}\chi_{A\setminus K_{\epsilon}}(x)dx < \epsilon.$$ ...
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1answer
70 views

Finding the integral $\int_0^\infty\sin(x^n)dx$ [duplicate]

Is there a non-complex number involving method to find the following integral $$\int_0^\infty\sin(x^n)dx$$ Maybe something using the idea similar to that for evaluating $\frac{\sin x}{x}$ under ...
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0answers
19 views

Triple integral : volume of $4x^2+3y^2=z^2+2$

I need to find the volume of this region $4x^2+3y^2=z^2+2$ for $|z|\le 2$ . It's an elliptic hyperboloid . $4x^2+3y^2=z^2+2=\sigma$ $\frac{x^2}{\sigma/4}+\frac{y^2}{\sigma/3}=1$ volume = $\...
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2answers
203 views

Having trouble computing $\int_3^5\frac{t}{1+0.1t} dt $

$$\int_3^5\frac{t}{1+0.1t} dt $$ For some reason this is equal to: $$\frac{1}{0.1\left(2 - \left(\frac{\frac{1}{0.1}}{\ln1.5 - \ln1.3}\right)\right)}$$ I have no idea how to reduce to that.
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0answers
14 views

An integral involving a Gaussian, error functions and the Owen's T function.

This question is closely related to An integral involving a Gaussian and an Owen's T function. and An integral involving error functions and a Gaussian . Let $\nu_1 \ge 1$ and $\nu_2 \ge 1$ be ...
8
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2answers
89 views

Integrate $\frac{1}{x\,\log{x}}$ by parts

A naive indefinite integration of the function $\dfrac{1}{x\,\log{x}}$ can be performed as follows: Let $ \begin{eqnarray} I &=& \int\dfrac{dx}{x\,\log{x}}\\ \therefore I &=& \...
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1answer
31 views

Integrating continuous function of two variables by one of the variables, do I get continuous function?

Let $f(x,y)$ be a function continuous in $x$ such that $g(x) = \int_a^b f(x,y) dy$ exists for every $x$. Is $g(x)$ necessarily continuous? I am especially interested in the Riemann and Lebesgue ...
0
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3answers
54 views

Integration, get stuck at x=tan($\theta$) when calculating arc length

I am learning to calculate the arc length by reading a textbook, and there is a question However, I get stuck at calculating $$\int^{\arctan{\sqrt15}}_{\arctan{\sqrt3}} \frac{\sec{(\theta)} (1+\tan^...
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3answers
45 views

Cancelling out constants with u-substitution integration

When using u-substitution to integrate, I tend to think about adding constants to make my $dx$ match my $du$. I don't have a basic enough understanding to grasp why it won't work if my $du$ contains ...
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1answer
47 views

How to show Riemann integrability

Is the function $$f(x) = \begin{cases} \frac{x^2}{2}+4 &, x\ge0 \\ \> \frac{-x^2}{2}+2 &, x<0.\end{cases}$$ Riemann integrable in the interval$[-1,2]$? Does there ...
3
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2answers
133 views

Confusion with integrating sin(nx)sin(mx) and Kroenecker delta

The specific integral I'm working with is the following: $$ \int_0^a\sin(n\pi y/a)\sin(n'\pi y/a) $$ This is supposed to come out to $0$ in the case that $ n \neq n' $ and $\frac{1}{...
0
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1answer
42 views

Surface Integral of Vector Field

Given the scalar field $$\phi(\vec{r})=\frac{1}{|\vec{r}-\vec{a}|},$$ where $\vec{a}=(-2,0,0)$, and the corresponding vector field $$\vec{F}(\vec{r})=\operatorname{grad}\phi,$$ as well as the surface $...
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1answer
21 views

Find integral of this fraction with a radical. Is my process right?

I think this involves a u-sub and then partial fractions? Is this process right? Basically I learned any integral can be solved with using usub and integration by parts... and using partial fractions ...
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0answers
19 views

If $f:[a,b]\to\mathbb{R}^2$ ($n>1$) is a continuous rectifiable path, then m(f([a,b]))=0 [duplicate]

If $f:[a,b]\to\mathbb{R}^2$ is a continuous rectifiable path, prove that the measure of $f([a,b])$ is null. I've tried to use the fact that $f$ is uniform continuous, but I only got that $m(f([a,b]...
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2answers
25 views

Question about a Bernoulli's integration.

This is taken from bernoulli's book HYDRODYNAMICS chapter X, p.258. My question is about the integration. Can it be correct or there is an error?
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1answer
19 views

Multiple Integrals with simulation [on hold]

Aproximate the next integral (using a spreedsheet) with n=1000 and 5000, provinding a confidence interval of 95% to the aprox. Also, calculate the length of the confidence interval enter image ...
2
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0answers
59 views

Special Functions defined by Integrals.

There was this integral which caught my attention, when I was checking out The Applications of Beta and Gamma Functions. So, how can i prove the below, using change of variable? $$\int_{0}^{1}\frac{...
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1answer
49 views

How to calculate the integral of exponential functions?

Having an integral like $\int_{2}^{10}{\frac{x}{\ln x}}dx$ How does this function turns to an exponential integral of the form: $ \operatorname{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}t\,dt.\,$ For ...
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1answer
74 views

How do I calculate?

How do I calculate cross-sectional area of this: ${r}=\sqrt{sin\theta} \, $, when $ 0 \le \theta \le \pi,$ Don't know what is the right answer but I have get that the area is 1. Is that right answer?...
4
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1answer
42 views

Trying to understand a standard example of weakly but not strongly measurable function

I consider quite a standard example of a function that is weakly measurable but is not strongly measurable. Unfortunatelly, I don't fully understand it. Let $X=l_2([0,1])$. Then $X$ equipped with a ...
1
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4answers
51 views

How to integral: $\int x^{2}e^{-\frac{x}{2}}dx$?

How to integral: $\int x^{2}e^{-\frac{x}{2}}dx$ ? I try to use the rule: $\int udv=uv-\int vdu$, and $u=x^{2}, du=2xdu, dv=e^{-\frac{x}{2}}dx, v=-2e^{-\frac{x}{2}}$, and $\int x^{2}e^{-\frac{x}{2}...
9
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1answer
70 views

Polygon on a grid

Given a square constructed on a grid of points with integer coordinates, what is its maximum area, if we know that there are exactly 3 grid points in its interior? I have no idea how to start. I ...
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1answer
30 views

How do I calculate area?

How do I calculate the area of this: $D=\{ (x,y)\mid 0 \le x \le 1, x^2 \le y \le x^2+2 \}$ ${A}=\iint_D \, \textrm{d}A.$ Don't know what is the right answer but I have get that the area is 6. Is ...
1
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1answer
68 views

Integral $\int\limits_{0}^{\infty}e^{-\frac{1}{2}\left(y^2+\frac{t^2}{y^2}\right)}\,dy$

I have no idea ho to compute this integral: $$\int\limits_{0}^{\infty}e^{\large-\frac{1}{2}\left(y^2+\frac{t^2}{y^2}\right)}\,dy$$ I have put this integral into Wolfram Mathematica and the result ...
4
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4answers
161 views

Tricky integral?

I tried to calculate this integral: $$\int_0^{\frac{\pi}{2}}\arccos(\sin(x))dx$$ and my result was $\dfrac{{\pi}^2}{8}$, but actually, according to https://www.integral-calculator.com/, the answer is $...
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0answers
23 views

Define a measure from density function with symmetry

A little bit similar to what I have asked before: Change variable in the integral with nonnegative measure Let $f_X(x)$ be a probability density function with $x\in \mathbb{R}^n$ and $X\subset\...
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0answers
28 views

Alternative proofs for Leibniz Rule

Let $f: [a,b] \times [c,d] \to \mathbb{R}$ be a continuous function with $U \subset \mathbb{R}^{n}$ and $\frac{\partial f}{\partial y}$ continuous. Take $F(x,y) = \int_{a}^{b}f(x,y)dx$ and show that ...
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1answer
13 views

Joint to marginal PDF, bounds of integration? (Bertsekas, Tsitsiklis, Question 3.4.15)

I am working on Question 3.4.15 from the second edition of Introduction to probability by Bertsekas and Tsitsiklis: "A point is chosen at random (according to a uniform PDF) within a semicircle ...
1
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1answer
46 views

Why is this integral equality true? $\int\limits_{-\pi}^{\pi}|a+be^{it}|dt=\int\limits_{-\pi}^{\pi}\left||a|+|b|e^{it}\right|dt$

I was reading an article, and while proving a proposition, they state that if $a,b\in\mathbb{C}$, then due to periodicity we have $$\int\limits_{-\pi}^{\pi}|a+be^{it}|dt=\int\limits_{-\pi}^{\pi}\left||...
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0answers
31 views

Finding a multivariable function using it's antiderivative

If I'm given: $\frac{f(x,y) - f(x-y,y)}{y} = g(x); \forall y \in \mathbb{R}$. Can I do the following steps: $$ \lim_{y\to0} \frac{f(x,y) - f(x-y,y)}{y} = g(x)$$ $$ \lim_{y\to0} \frac{\partial{f(x,y)}}{...
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0answers
35 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
3
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4answers
83 views

Limit of $e^{x^2}\int_{x}^{+\infty} e^{-t^2}dt$ [on hold]

I'd like to find this limit : $\lim_{x\to +\infty}{e^{x^2}\int_{x}^{+\infty} e^{-t^2}dt}$ . I can't use L'Hôspital's rule (and the error function), some help would be greatly appreciated
3
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2answers
59 views

Basic Integration Problem: $\int \frac{di}{i-V/R}$

This is my first time using the maths stack forum, I normally use the electrical engineering stack. I am having difficulty taking the integral of a term (see the following link). $$\int \frac{di}{i}=\...
2
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1answer
25 views

How to find limits of integral to find volume?

Find the volume generated by the plane region, in the first quadrant, bounded by the graph of the function $ y=\sqrt{9-x^2} $ sbout the y-axis. I know how to solve it using the formula but how do I ...
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1answer
21 views

Trying to find the CDF of $X+Y$ when $X\sim exp(\alpha)$ and $Y \sim exp(\beta)$ (independent) without convolution, but it doesn't seem to work

The textbook I am using, using convolution in order to find the CDF of the $X+Y$ when $X\sim exp(\alpha)$ $Y\sim exp(\beta)$, and X and Y are are independent. However, I have no background with ...
1
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4answers
52 views

How to integrate $\cos^3x$ by parts

I've converted $\cos^3(x)$ into $\cos^2(x)\cos(x)$ but still have not gotten the answer. The answer is $\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$ My answer was the same except I did not have a $3$ ...
1
vote
2answers
50 views

Integral of $\frac{1}{\sqrt x(1+16x^2)}$

I want to find the integral of $\frac{1}{\sqrt x(1+16x^2)}$ With the substitution $u=4\sqrt x$ I get $\int\frac{du}{2(1+u^4)}$ which is quite complicated to integrate. Is there an easier way to do ...