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
votes
1answer
41 views

What does the region enclosed by $y = 1/x^5 , y=0, x=3, x=4$ look like?

I'm having trouble trying to see what the region I'm supposed to be computing looks like. The volume of the solid obtained by rotating the region enclosed by $$y = 1/x^5 , y=0, x=3, x=4$$ about the ...
2
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0answers
65 views

Integration of $\int_a^b{f(x)^2 dx}$ for general $f(x)$?

I working on a math problem that requires an elegant, general solution for $$\int_a^b{f(x)^2\,\mathrm{d}x}.$$ I am struggling to find one and hope that somebody here can help out. By using integration ...
1
vote
2answers
46 views

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is integrable.

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is Riemann integrable on $[0,1]$. I think for everyone this question is really basic, but I'm just training myself on proving the ...
2
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0answers
44 views

How to evaluate given integral in terms of Gamma function?

I want to evaluate the below integral $$ \int_{0}^{\infty} e^{-ax^{2}-bx} x^{\alpha -1}\,\mathrm{d}x, ~~~~~~ \operatorname{Re}(\alpha)>0 $$ where $a,b$ are real constants. Somehow, I want to ...
1
vote
1answer
14 views

Integration of autocorrelation function when $\int_0^1 f(t) dt = 0$

When $\int_0^1 f(t) dt = 0$ and $f(t)=0$ for $t\in \mathbb{R} \setminus [0,1]$, my conjecture is that $$ \int_{\mathbb{R}} R_{ff}(\tau)\,d\tau = 0, $$ where $$ R_{ff}(\tau) = \int_{\mathbb{R}} f(t+\...
2
votes
3answers
82 views

Question regarding fixed point (integrals)

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^1 f(t)\,\mathrm{d}t =\frac{1}{2},$$ then $f$ has a fixed point in $[0,1]$, i.e. $\exists x_0\in [0,1]$ such ...
3
votes
1answer
81 views

Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$

Let $f(x)=\cos^2(x)$ if $x\in \mathbb{Q}$ and $f(x)=0$ if not. Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$. The problem that I have, is that I don't really see why this function ...
0
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1answer
38 views

Determine the convergence or divergence of the sequence with $n$th term $n\pm(-1)^n$ [closed]

So I was given the following prompt when I was studying for a test: In the following sequence, determine the convergence or divergence with the given $n$th term. $\lim_{n \to \infty} n\pm(-1)^n$ I ...
1
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3answers
42 views

Proving that the $\chi^{2}$ Distribution is a PDF

The $\chi^{2}$-distribution is given by $$f(x; k)\equiv \frac{1}{2^{\frac{k}{2}} \Gamma\left(\frac{k}{2}\right)}x^{\frac{k}{2}- 1}e^{-\frac{x}{2}}.$$ If the $\chi^{2}$-distribution is a probability-...
0
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0answers
16 views

Integral over factorials/reciprocal beta function / generatnig function of $\ln\left({\ln(1+c)\over c}\right)$

I know from the product representation of $\ln(1+c)/c$ that the following: $$\ln\left({\ln(1+c)\over c}\right)=\sum_{n=1}^{\infty}\int_{-1}^0 \frac{(n+x)!}{n! x! n}c^n (-1)^n dx $$ I wondered what ...
0
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1answer
59 views

How can this integral be evaluated?

I am trying to find the definite integral of this function. When entered into wolfram alpha the result (shown below) is given. However I do not understand what the E (elliptic integral of the second ...
0
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0answers
31 views

Integrating Bessel's Function [closed]

Here a last year student, trying to program an analytical model for borehole heat exchangers. I have been researching for weeks on how to get a discrete solution of these equations, so that i can ...
-3
votes
0answers
49 views

How should I calculate the following integral? [closed]

How should I calculate the following integral? $$\int_{-c}^{+c} \frac{\exp\left({-2\beta (a^2+y^2)^{1/2}}\right)}{\sqrt{a^2+y^2}}dy$$ $a$ and $\beta$ are constants and $[-c,+c]$ is the custom range.
1
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1answer
43 views

The convergence of the series after integrating term-by-term

I am interested in the integral $$ I = \int_{-\infty}^\infty e^{i A x} e^{- Bx^2 - \lambda x^4}, $$ for some real values $A$, $B>0$ and $\lambda >0$. One way to tackle this integral is to expand ...
-1
votes
0answers
30 views

Evaluating integral $\int \frac{2x+7}{2x^2+x+3} \,dx$ [closed]

$$\int \frac{2x+7}{2x^2+x+3} \,dx$$ Any hints on how to begin?
0
votes
0answers
30 views

$\int_0^x \frac{t^n}{1-t} dt = ?$ [duplicate]

I want to calculate the integral, $\int_0^x \frac{t^n}{1-t} dt$, where $n \in \mathbb{N}$ and $x \in (0;1)$ are some constants, but I do not know how to do it. I tried the following methods: ...
0
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0answers
16 views

Is it possible to express an infinitesimal area as a sum of squares?

I came across a question which was asking the center of mass of an area, the centroid. The proposed solution uses weighted average, the formula is that given a planar region, the x and y coordinates ...
0
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0answers
17 views
2
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1answer
44 views

Compute $\int_{\gamma} z\, dz$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$

Compute the complex line integral$$\int\limits_{\gamma} z\, \mathrm{d}z$$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$. Confused as to how I am supposed to go about ...
2
votes
2answers
62 views

Find $\int_0^4(g\circ f\circ g)(x)\mathrm{d}x$ where $f(x)=\sqrt[3]{x+\sqrt{x^2+1/27}}+\sqrt[3]{x-\sqrt{x^2+1/27}}$, $g(x)=x^3+x+1$

Let $$f(x)=\sqrt[3]{x+\sqrt{x^2+\frac{1}{27}}}+\sqrt[3]{x-\sqrt{x^2+\frac{1}{27}}}$$ and $$g(x)=x^3+x+1$$ then, find $$\int_0^4(g\circ f\circ g)(x) \mathrm dx$$ My attempt: Let $\displaystyle h(x)=\...
1
vote
1answer
43 views

Integral of $\int\ln(a\sin(2x)+b)\cos(x)dx$

Mathematica is able to solve the indefinite integral $$\int\ln(a\sin(2x)+b)\cos(x)dx$$ analytically for $b>a>0$, but the resulting $\arctan$ terms are too complicated for the next steps I need ...
1
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2answers
72 views

What is the other way to write this triple integral?

I need to find all $5$ ways to write this integral and so far I have found $3$ of the $5$ but I need some help to find the last $2$ which are $dy$ $dz$ $dx$ and $dy$ $dx$ $dz$. I assume one of them ...
0
votes
1answer
26 views

Could this Beta density related integral be simplified

I encounter the following integral, where $F^{-1}(u)$ is the quantile function for $\mathcal{N}(0,1)$ distribution: for $k \in\mathbb{N}$: $$\int\limits_{1/2}^1 F^{-1}(u) u^k(1-u)^k\,\mathrm{d}u$$ ...
13
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2answers
340 views
+100

How to Evaluate $\int_{0}^{1} \frac{x^2 \ln(1-x^4)}{1+x^4}$?

How to evaluate $$ \int_{0}^{1} \frac{x^2 \ln(1-x^4)}{1+x^4} \,dx \approx -0.162858 \tag{1}$$ The integral arises in the computation of $$\left( \sum_{n=1}^{\infty} \frac{(-1)^n}{n}\right)\left(\sum_{...
2
votes
1answer
20 views

Finding initial velocity with only acceleration and distance

I’m having a hard time trying to figure this question out. With what initial velocity must an object be thrown upward (from ground level) to reach the top of the Washington Monument, which has a ...
0
votes
2answers
33 views

Let $a>R>0$. Calculate the volume of the body that meets the differences $x^2+y^2+z^2 \leq a^2$ and $x^2+y^2 \leq R^2$

Let $a > R > 0$. Calculate the volume of the body constricted by $x^2+y^2+z^2 \leq a^2$ and $x^2+y^2 \leq R^2$ I don't know where to start with this problem. The constraints form a cylinder ...
1
vote
3answers
51 views

Need help to solve ODE $y'(x) = -\tanh(y(x))$

I want to solve the ODE $$y'(x) = -\tanh(y(x))$$ It is separable, so diving by $\tanh(y(x))$ and integrating wrt $x$ we get $$\int \frac{y'(x)}{\tanh(y(x))} \,dx = \int -1\,dx$$ I am having trouble ...
0
votes
0answers
37 views

Construction and integration of differential forms

I'm computing measures on quotient spaces of SO(3) and I have questions about the integration of these measures. I'm parameterizing SO(3) by the Euler angles $R(\phi,\theta,\psi) = R_z(\phi)R_y(\theta)...
1
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0answers
34 views

Why does $\int\limits_{0}^{\infty}P(A_{\infty}\geq c^{2})dc=E[\int\limits_{0}^{\sqrt{A_{\infty}}}dc+\int\limits_{0}^{\sqrt{A_{\infty}}}dc]$

Why does $\int\limits_{0}^{\infty}P(A_{\infty}\geq c^{2})dc=E[\int\limits_{0}^{\sqrt{A_{\infty}}}dc+\int\limits_{0}^{\sqrt{A_{\infty}}}dc]$? My attempt: $A_{\infty}\geq c^{2}\iff \sqrt{A_{\infty}} \...
0
votes
2answers
77 views

Integrability of Thomae function and value of its integral

I am trying to solve the following exercise (from Axler's Measure, Integration and Real Analysis) and I would like to have some help in finishing my proof. "Define $f:[0,1]\to\mathbb{R}$ as ...
3
votes
1answer
100 views

Show that $f$ is Riemann-Integrable

Let $f$ be a function defined on $[0,1]$ by: $f(x)=1$ if $x=\frac{1}{n}, n \in \mathbf{N}^*$ and $f(x)=0$ if not. Show that $f$ is Riemann integrable on $[0,1]$. I know that there is already a post on ...
1
vote
1answer
60 views

Why is the integral between $\cos(x)$ and $\sin(x)$ between the intersections $\sqrt{2}$

I just recently noticed, that the integral $$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos(x) - \sin(x) dx = \sqrt{2}.$$ I know who to calculate all of it but I was quite surprised that the result is so ...
0
votes
1answer
28 views

How to verify the funtion $\Bbb{P}_{a}$ , defined as $\Bbb{P}_{a}(B) = \int_{B} f d\lambda$ for any Borel set B, is a probability measure

I am trying some exercise on a measure theoretic probability text, and want to make sure if I am doing right. The question is: Let ($\Omega$, $\mathbf{F}$ , $\Bbb{P}$) be a probability space. Assume g ...
0
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0answers
15 views

correlated joint probability calculation [closed]

I was wondering how to calculate this joint probability , $P(x>\max (a,y), a<y <b)$, where $a$ and $b$ are constants and $x, y$ are variables
0
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0answers
18 views

Integral of time correaltion function

I am reading about time correlation functions. And I am stuck with the following integral $$ \langle |\mathbf{r}(t)-\mathbf{r}(0)|^{2} \rangle = 6 \int_{0}^{t}\left[\int_{0}^{t^{'}}Z(t^{'}-t^{''})dt^{...
-1
votes
1answer
53 views

using reduction formula evaluate $\int_0^a (a^2 + x^2)^{5/2} dx $ [closed]

Obtain reduction formula of the expression $$\int_0^a (a^2 + x^2)^{5/2} dx $$
0
votes
1answer
27 views

Rate of convergence to zero of an integrand given its integral is convergent?

Let $h(x)$ be real-valued, monotonically-decreasing function on the non-negative reals such that $\int_{0}^{\infty} x^{k+1} h(x) \, dx$ is convergent for some positive integer $k$. Does it follow that ...
-1
votes
1answer
160 views

Solve $\int_{0}^{\pi/2} \arccos\left( \frac{\cos(x)}{1+2\cos(x)} \right) \mathrm dx$

$$\int_{0}^{\pi/2} \cos^{-1}\left( \dfrac{\cos(x)}{1+2\cos(x)} \right) \,dx$$ The final answer is: $\dfrac{5\pi^2}{24}$
0
votes
1answer
21 views

Convergence of integral implies integrand is asymptotically bounded above by a power law that has a convergent integral?

Let $f$ be a continuous, real-valued function on $[1,\infty)$ such that $\int_{1}^{\infty} f(x) \, dx$ is finite. Does this necessarily imply that there exists $\kappa > 0$ such that $f(x) = O\!\...
1
vote
2answers
71 views

Can $ \int \sin(x)+\sum^{\infty}_{n=1} (-1)^n\frac{\sin^{(2n+1)}(x)}{(2n+1)!} dx$ be evaluated in terms of elementary functions?

How do you go about integrating this and can it even be done with elementary functions? $$ \int \sin(x)+\sum^{\infty}_{n=1} (-1)^n\frac{\sin^{(2n+1)}(x)}{(2n+1)!} dx. $$ I understand the concept of $$ ...
0
votes
2answers
48 views

How do I evaluate $\int_{0}^{\sqrt{2}}\int_{x}^{\sqrt{4-x^2}}{\sqrt{x^2+y^2}} \, dy \, dx$?

I'm having troubles evaluating this double integral. Can somebody help me? I've gone to the part that I need to use trigonometric substitution, but performing the said sub, I think I'm kind of unsure ...
1
vote
1answer
44 views

How to prove that $\int_{[-\pi,\pi]}\log(\vert 1- \exp(it)\vert)\mathrm{d}\lambda(t)=0$?

Let $r>1$ and $\int_{[-\pi,\pi]}\log(\vert 1- r\exp(it)\vert)\mathrm{d}\lambda(t)= 2\pi\log(r)$. We want to prove that : $\lim \limits_{r\to 1} \int_{[-\pi,\pi]}\log(\vert 1- r\exp(it)\vert)\mathrm{...
0
votes
1answer
36 views

How to find critical points of definite integral

Say I have a function $$g(x) = \int_a^b (f(t)-x)^3dt$$ how would I go about finding the critical points of this function? I know that FTC gives you that if $$h(x) = \int_0^x f(t)dt$$ then $$h'(x) = f(...
0
votes
2answers
35 views

How to solve for the function $f(x)$ such that the area underneath is always equal to the arclength?

First, I set up my equation. The area is obviously $\int f(x) \space dx$. The arclength is $\int \sqrt{1+(\frac{dy}{dx})^2} \space dx$. Thus, I have the equation $\int f(x) \space dx = \int \sqrt{1+(\...
0
votes
0answers
33 views

How would I evaluate this limit using summation equations?

This is an alternate way of finding the area under a curve for a certain function: $$\lim_{n \to \infty} \sum_{k=1}^n \frac{3}{n} \sqrt{9-\left(\frac{3k}{n}\right)^2}$$ This expression would give: $$\...
1
vote
0answers
18 views

Proove $||\xi||^\alpha =\int_{\mathbb{R}^n} (\cos(\xi y) - 1) {||y||^{-(n+\alpha)}} \: dy$

I came a across a problem I can't solve. Given $n\in\mathbb{N}$ and $\alpha\in (0,2)$, proove: $||\xi||^\alpha =\int_{\mathbb{R}^n} (\cos(\xi y^T) - 1) {||y||^{-(n+\alpha)}} \: dy$ for any $\xi,y\in\...
0
votes
0answers
28 views

How to solve the integral of a step function

Here is explained how a step function (goes from 0 to 1 at t = 0) is related to a ramp function (see this link for detail https://en.wikibooks.org/wiki/Control_Systems/System_Metrics) I do not ...
0
votes
2answers
34 views

How to write the equation of an off-center circle in terms of r (parameterization)

I've looked around online for a while and I can't seem to find a question that properly captures what I'm talking about (or answers that really help me). For a homework problem, I have to find the ...
1
vote
0answers
38 views

Evaluating an integral involving Beta function ratio

Does anyone have an effective strategy to evaluate the following integral? Is it perhaps some known special function? \begin{align} & \int_0^1 p^x (1-p)^{n-x} \frac{B\left( y+sp, m-y+s-sp \...
0
votes
1answer
61 views

Integrating summation $\int \sum_{n=0}^\infty \sin^n x dx$ [closed]

Just to give a base example $$\sum_{n=0}^\infty \sin^n x$$ What are the rules for going about integrating a sum equation where $n$ is the power of the $\sin(x)$?