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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0
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1answer
27 views

Difference between two integrals with PDFs

I am trying to evaluate $$\int_{E'}^{\bar{v}}(v-c)g(v)dv-\int_{E'}^{\bar{v}}(v-E')g(v)dv$$ where $v$ is a random variable with support $[\underline{v}, \bar{v}]$, and $c<E'$ (both of which are in $[...
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1answer
52 views

Inverse trigonometric substitution for integrals

I am trying to solve this integral by inverse trigonometric substitution. $$ \int_{0}^1{\sqrt{4kx-k^2x^2}dx} $$ where k is an arbitrary constant. I completed the square to get this:$$ \int_{0}^1\sqrt{...
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0answers
21 views

Simplifying an integral with a pdf

I have $$\int_{E'}^{\bar{E}}(v-c)g(v)dv-\int_{E'}^{\bar{E}}(v-E')g(v)dv$$ where $v$ is a random variable distributed on $[v\in{\underline{v},\bar{v}}]$ with pdf $g(v)$, $c<E'<\bar{E}$, and all ...
2
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0answers
54 views

I have confirmed that the left integral diverges by the special value of the Lambert function in the equation $xe^{x^{2}}=1,$ anymore of it ??

Problem. Prove that $$\int_{0}^{\infty}\frac{{\rm d}x}{x^{2}e^{x^{2}}}= \infty$$ Indeed, that's true because $$\int_{0}^{\infty}xe^{x^{2}}{\rm d}x= 1/2\int_{0}^{\infty}{\left ( x^{2} \right )}'e^{x^{...
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1answer
36 views

Integrability of increasing function

I am trying to prove the following (from Axler's Measure, Integration and Real Analysis): "Suppose $f:[a,b]\to\mathbb{R}$ is an increasing function (i.e. $c,d\in [a,b], c<d\Rightarrow f(c)\leq ...
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3answers
42 views

Simple Integration of Exponential Function

I would like to integrate $3.5 \times 10 ^{-8} \times 2.737^{x}$ Any help is appreciated.
4
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1answer
37 views

Total work compressing a conical spring

A conical spring's stiffness varies linearly with displacement from rest. Its stiffness when uncompressed is 45 N/m, and 150 N/m when fully compressed. The uncompressed spring stretches 30 cm further ...
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5answers
100 views

Use integration by parts twice $\int_{0}^{\frac{\pi}{8}}x^2\cos(4x)dx$ [closed]

I am confuse about this integration by two parts I don't get the steps and any help would be much appreciated i know the answer is 0.00730 but don't know how to get to it. This is the work
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1answer
124 views

What did I do wrong with this integral problem?

Find the value of the following definite integral $$\int_0^{\pi/2} \sqrt{1 - \sqrt{3}\sin 2x + 2\cos^2x} \, dx$$ My attempt: $$\int_0^{\pi/2} \sqrt{1 - \sqrt{3}\sin 2x + 2\cos^2x} \, dx$$ $$= \int_0^...
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1answer
45 views

Comparison Test for $-\ln\left(1-\dfrac{\pi^2}{2x^2}\right)$

I'm trying to use the comparison test to show that $\displaystyle \int_3^\infty -\ln(1-\frac{\pi^2}{2x^2})$ converges, but I'm having trouble finding a function that is greater than it and still ...
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1answer
67 views

How to evaluate the integral $\int_{-\infty}^a e^{-x^2} dx$

How to evaluate the integral $\int_{-\infty}^a e^{-x^2} dx$? I know that $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$ because if I let $I = \int_{-\infty}^{\infty} e^{-x^2} dx$, then \begin{...
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0answers
60 views

Asymptotic evaluation from Bender and Orszag

In Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers, Exercise 6.56 (c) asks to find the leading behaviour of the integral $ \int_0^1 \cos(x t^4)\tan t\;\mathrm{d}t$ as $x\...
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1answer
71 views

$\int\sqrt{\sin(x)}(\cos(x))^3dx$

EDIT: @Andrew Chin's comment solved the issue $\int(\sin(x))^{1/2}(\cos(x))^3dx$ =$\int(\sin(x))^{1/2}(\cos(x))^2(\cos(x))dx$ =$\int(\sin(x))^{1/2}(1-\sin^2(x))(\cos(x))dx$ Let $u=\sin(x)$, therefore $...
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2answers
62 views

How to compute a complex integral of $x \: dz$?

I'm working through chapter 4 of Alfohrs, and the first exercise is computing the integral $\int_\gamma x \: dz$ where $\gamma$ is the directed line segment from $0$ to $1+i$. What, exactly, does this ...
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1answer
39 views

discrete sum greater than integration

Why for any $j \in N$ (the natural numbers) we have $\sum_{n = j+1}^{j^2} \frac{1}{n} \geq \int_{j+1}^{j^2 + 1} \frac{1}{x} dx $
4
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3answers
148 views

A different way to calculate $\int_0^\infty \tfrac{1}{1+x^n} dx$.

Most of you might have stumbled upon this integral, when studying complex analysis, namely via contour integration, but today I asked myself if there is a different way to calculate it. I'm almost ...
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0answers
57 views

How many points of integration are necessary to approx $\int_0^1{e^{e^{-x}}}dx$ with 4 correct decimal places using the trapezoid rule?

How many points of integration are necessary to approx $\int_0^1{e^{e^{-x}}}dx$ with 4 correct decimal places using the trapezoid rule? I know that what I need here is $$|e_r| \le 5\times 10^{-(m+1)},...
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1answer
37 views

What to set $u$ as in integration by substitution

$$\int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}}x^2\sqrt{1-x^2}\ \mathrm{d}x$$ The question asks me to use a trigonometric substitution so I tried $u=\sin x$ but that doesn't give me nice $x$ limit values ...
4
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2answers
95 views

$ \int_c^\infty \frac{e^{-a x^2 + bx}}{x} \, dx $

Suppose $a > 0$, $b$ is an arbitrary real number, and $c$ is a positive real number. Is there any closed form solution for the following integral $$ \int_c^\infty \frac{e^{-a x^2 + bx}}{x} \, dx \...
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2answers
75 views

What is the antiderivative of one over an arbitrary polynomial?

Suppose we have an arbitrary polynomial as follows. $$p(x):=\sum_{i=0}^n a_nx^n$$ And we want to find the antiderivative of $(p(x))^{-1}$ with respect to $x$. $$\int\frac{dx}{p(x)}$$ Messing around ...
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4answers
81 views

How can we simplify this integral? $\int{x\frac{f(x)}{\int f(x) dx} dx}$

In a machine learning lecture, we encountered the following integral that we needed to solve to calculate the mean of some random variable $x$: \begin{equation*} \int{x\frac{f(x)}{\int f(x) dx} dx} \...
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1answer
56 views

Calculate $\iint_D \frac{\sqrt{x^2 + y^2}}{1+x^2+y^2} dx\,dy$

I am attempting to solve the integral: $$\iint_D \frac{\sqrt{x^2 + y^2}}{1+x^2+y^2} dx\,dy\,,$$ where $D$ is bounded by $1<x^2+y^2<9$ and within the sector bounded by lines $$-\frac{y\sqrt{2}}{2}...
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0answers
29 views

2D Fourier Transform of a function?

Here I have a function like: $\phi(x,0,z) = \int_0^\infty\int_0^\infty A_1\sin(ux)\sin(vy)dudv$ and I would like to find the form of $A_1$ which I have been told is the 2d Fourier Transform of $\phi(x,...
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0answers
28 views

$f: [0, 1] \to \mathbb{R}$ is continuous. Prove existence of $c \in (0, 1)$ where $\int_{0}^{1} f(x) x^2 dx = \frac{f(c)}{3}$ [duplicate]

My approach with partial integration: $$\int_{0}^{1} f(x) x^2 dx = \frac{f(1)}{3} - \int_{0}^{1} f'(x) x^2 dx = \frac{f(c)}{3}$$ By mean value theorems, there exists $d \in [0, 1]$ such that: $$\int_{...
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0answers
62 views

Calculating $p$-norms for $f_\epsilon(x)=x^{-1/p+\epsilon}\chi_{(0,a)}(x)$

I have some problems calculating this norm, I can't get the integral right. I have the following function $f_{\epsilon}(x)= x^{(-1/p)+\epsilon}\mathcal{X}_{(0,a)}(x)$ with $p>1$, $a>0$ and $0&...
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0answers
49 views

Need to solve an integral [closed]

I need help to solve the following integral $\int_{0}^{\infty}x^n \exp(-ax^m-bx^2)\,dx$ Any help will be appreciated.
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1answer
25 views

Using the surface of the unit sphere to calculate $\int_{\mathbb R^3}\chi_{\{x_3>0\}}\chi_{B^c_{1/n}(0)}\exp(-\lvert x\rvert^2)/\lvert x\rvert dx$

I'm struggling with finding a proof idea for this integral from an integration theory practice exam. Suppose $$f_n:=\frac{\exp(-\lVert x\rVert_2^2)}{\lVert x\rVert_2}\chi_{\{x_3>0\}}(x)\chi_{B^c_{1/...
6
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2answers
106 views

Convergence of integral and summation for Time taken for complete revolution around vertical circle

I want to find Time taken to complete Vertical circular motion by Particle of mass $m$ So I proceed as follows Applying work energy theorem $$-mgR(1-\cos(\theta)=\frac12 mv^2-\frac12 mu^2$$ $$\implies ...
0
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2answers
73 views

Solving integral involving Dirac delta function i.e$\int_{-\infty}^\infty e ^{\delta(x)} dx$

I know some properties of Dirac-delta function specially those which are useful for integration involving these functions such as $$\int\limits_{-\infty}^{\infty}\delta(x)\,\mathrm{d}x$$ But I am ...
1
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1answer
32 views

Euler's Beta function for positive integers derivation

I am aware that a proof for the Euler's Beta function for positive integers has already been asked, but I have tried to derive it myself without using the gamma function, and my result doesn't match. ...
0
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0answers
30 views

What tools can we use to visualise volume integrations?

What tools do people use to visualize integrated volumes? I'm horrible at drawing perspectives, especially with hollowed-out 3D graphs. Can't imagine how to do that going beyond the usual three ...
2
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2answers
78 views

Integral involving exponentials of cosh functions

I recently encountered the following integral $$I_p\equiv \int\limits_0^{+\infty}\,\int\limits_{-\infty}^{+\infty}\,e^{-p\,[c+(c^2+1)\cosh x]}\,\mathrm{d}x\,\mathrm{d}c\,,$$ with $p>0$. Can $I_p$ ...
0
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0answers
30 views

Integral approximation (Laplace's method)

in order to show a result relating to Laplace's method of approximating integrals, I am stuck with the following exercise. Let $f,g\in C(I,\mathbb R)$ where $I$ is a compact interval whose interior ...
1
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1answer
72 views

Area: Shell of a simple spiral [closed]

Staying inside the math of $2$D Polar coordinates, up to but may not include formulas for areas. The general equation is $r=f(θ)$ for many types of spirals. A point in polar coordinates is $(r,θ)$. ...
0
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1answer
99 views

Convergence integral Riemann zeta

In the paper of Riemann and the book of Edwards I encountered the following representation for $\zeta$: $$ \int_0^\infty\frac{x^{s-1}}{e^x-1}\,dx=\Pi(s-1)\zeta(s) $$ which due to the fast growth of $e^...
6
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1answer
143 views

Prove $\int_0^1 f(g(x)) \, dx \leq \int_0^1 f(x)\,dx + \int_0^1 g(x)\,dx$

$f$ and $g$ are continuous functions on $[0, 1]$. $f$ doesn't decrease on this segment. Values of both $f$ and $g$ are in range $[0, 1]$. Prove following inequality $$\int_0^1 f(g(x)) \, dx \leq \...
1
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3answers
60 views

Integrate $\int_2^5 \dfrac{1}{1+\sqrt{x-1}}\, \mathrm{d}x $ using substitution with a fraction

Im doing this question currently $$\int_2^5 \dfrac{1}{1+\sqrt{x-1}}\, \mathrm{d}x $$ and I set $u = x-1$ to get $\tfrac{\mathrm{d}u}{\mathrm{d}x} =1$ so $du =dx$. I then write it out as $$\frac{1}{1+\...
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0answers
24 views

Ratio of Definite Integrals

I have ended up with the following ratio of definite integrals: $$g(z)=\left[{\int\limits_0^1 f(x)e^{-zf(x)}dx}\right]^{-1}{\int\limits_0^1 f^2(x)e^{-zf(x)}dx}$$ where $f(x)$ is an unknown function. ...
0
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0answers
25 views

Counterexample to: $f$ is Riemann Integrable if and only if its discontinuous set is Jordan measure zero

The forward direction is covered in some textbooks. But the reverse I am only able to find sparingly. Is this a counterexample?: $f=\begin{cases} 0 & (x\not\in\mathbb{Q})\\ 1/n & (x\in\mathbb{...
2
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3answers
61 views

Integration with trig substitution

Trying to evaluate this using trig substitution: $$\int \frac {1}{49x^2 + 25}\mathrm{d}x $$ Here's how I'm going about it, using $x = 5/7(\tan\theta)$ $$\int \frac {1}{49\left(x^2 + \frac{25}{49}\...
-1
votes
1answer
34 views

For $f(t)=t^2-4t+3$ find $g(x)=\min_{(x\le t\le x+1)} f(t)$ and a value depending on $g$ [closed]

Let $f:\mathbb{R}\to\mathbb{R}$ $ ,f(t)=t^2-4t+3$ and $g(x)=\min_{(x\le t\le x+1)} f(t)$ . Find $$a=\frac{1}{ g'(1/2)+g'(3/2)}\int\limits^3_0 g(x)\,\mathrm{d}x$$ I had this problem on a test today and ...
0
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1answer
50 views

Unable to solve definite integral if an implicit function

We were asked to find the area under the given function, as a challenging exercise: \begin{gather*} 6x^{3} +11x^{2} y+6xy^{2} +y^{3} =x \end{gather*} The first thing I tried to do, was probably naive, ...
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2answers
31 views

Definite Integral from or to; plus or minus infinity

I always knew integrating in a common way: $\int_{n}^{z}{x^y} \ dx=[\frac{z^{y+1}}{y+1}] - [\frac{n^{y+1}}{y+1}]$ All we do is integrate $x^y$ and then replace the $x$ with $z$ first, then subtract it ...
3
votes
0answers
56 views

Integration by parts to form an infinite series

The integral $$I(\theta)=\int \sin(\theta)\frac{d\theta}{dt}dt$$ can be evaluated with the quotient rule by writing $I(\theta)=\int \sin(\theta)d\theta $. Instead, apply integration by parts more than ...
0
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0answers
17 views

Monotonic and non monotonic function's integrability(Riemann) question

I was told that all functions that have a finite number of discontinuities, are Riemann-integrable. Then, I found a proposition that tells the following: All monotonic functions on closed interval $[a,...
1
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2answers
110 views

Hard improper integral $\int_0^\infty \frac{(1+\frac{1}{x})^{-x}}{x^2}dx $

I have been stuck on this problem for a while: $$\int_0^\infty \frac{(1+\frac{1}{x})^{-x}}{x^2}dx $$ I have toyed around with substitution and integration by parts with no luck. I have also searched ...
-1
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0answers
29 views

$\frac{d}{dx} \int_{x_0}^{x_1} \rho u_{tt} dx$

In Partial Differential Equations by Walter Strauss, Ch 1.3 Example 2, the author differentiates the integral $$\int_{x_0}^{x_1} \rho u_{tt} dx$$ with respect to $x$ and finds that; $$\frac{d}{dx} \...
0
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2answers
53 views

Calculate polygon using some integrals

I have a polygon below. The length of it is $a$, base is $b$, and height is $h$. I want to have an integral to calculate volume of it. This is so different from what I know before, and it's not graph ...
1
vote
1answer
30 views

Argument $\int_{-\infty}^{t}\int_{-\infty}^{x_2}f_{X_1,X_2}dx_1dx_2=\int_{-\infty}^{t}\int_{-\infty}^{x_1}f_{X_1,X_2}dx_2dx_1$

Right now I have two integrals $$ \int_{-\infty}^{t} \int_{-\infty}^{x_2} f_{X_1,X_2}(x_1,x_2) \, dx_1 dx_2 = \int_{-\infty}^{t} \int_{-\infty}^{x_1} f_{X_1,X_2}(x_1,x_2) \, dx_2 dx_1 $$ with the ...
2
votes
3answers
135 views

Prove that : $ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{5}\zeta\left(3\right) $

Denoting $ \varphi=\frac{1+\sqrt{5}}{2}=\mathrm{Golden\ Ratio} $. How would you prove that : $$ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{...

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