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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22 views

How to deal with this integral equation?

While reading a paper I saw the following integral equation. $$\frac{1}{g} = \int_{-\pi}^{\pi} (\prod_{\sigma}\frac{\mathrm{d}p_{\sigma}}{2\pi}) \frac{\delta(\Sigma_{\sigma} ...
1
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3answers
42 views

improper integral for a function involving floor function

I am trying to study the improper integrability of $\int_{1}^{\infty}1-\frac{\lfloor x\rfloor}{x} d{x}$. I tried the definition of the improper integral as a limit with no success. Any hint?
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1answer
24 views

Lebesgue integration by parts in Sobolev space $W^{1,2}(\mathbb{R})$

Let $\phi, \psi \in W^{1,2}(\mathbb{R}) \subset L^2(\mathbb{R})$ and we want to integrate by parts the following piece: $$\int_{\mathbb{R}}\phi(x)\psi'(x)dx$$ Supposedly, it should look like this: ...
11
votes
1answer
98 views

To find the minimum of $\int_0^1 (f''(x))^2dx$

I was trying to solve a question of an entrance exam. I am completely stuck in the problem. I am not able to find idea how to proceed. Please help me. Let $A$ be the set of twice continuously di ...
2
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0answers
87 views

Triple integral with a change to spherical co-ordinates

I'm having trouble trying to figure out how to change the limits of integration to spherical co-ordinates in this particular question. I was wondering if someone would kindly be able to assist me in ...
0
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3answers
61 views

$\int_{-\infty}^{\infty}pe^{-tp^2} = 0$?

I want to show the following integral is equal to zero $$\int_{-\infty}^{\infty}pe^{-tp^2}$$ If I let $u = tp^2$, I get $dp = \frac{1}{2tp}du$ So I have ...
1
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1answer
57 views

Can someone show me if my steps are wrong in solving $\int \frac{\sqrt{1+\ln x}}{x \ln x} dx$

I just need to know if this method of solving this integral $$ \int \frac{\sqrt{1+\ln x}}{x \ln x} dx $$ is correct or not and if not where am I wrong?
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0answers
24 views

Applications of the Mean Value Theorem in Integration [on hold]

What are some interesting applications of the average value of a function when it comes to the topic of Integration? Thanks.
1
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1answer
24 views

Angular Velocity through integration

How to integrate $a\ddot\theta = \frac{gsin2\theta}{2} $ to find $r\dot\theta^2$ r is the radius. I am not sure how to integrate the equation with respect to t. Kindly explain.
2
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1answer
23 views

Integrals with complex functions: integration by parts and conjugate

I am working with integrals of complex functions. I assume all terms are well-defined. If $u=u(x,t):\mathbb{R}^n\times\mathbb{R}_+ \to \mathbb{R}$ (a real function), I have \begin{equation} ...
0
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1answer
102 views

For which values of $\alpha$ is $\int_{1}^{\infty}f(x)\, \mathrm{d}x$ convergent?

Let $\alpha>0$ and define $f(x)=\ln(x)/(x-1)^{\alpha}$ for all $x>1$. Before I got this problem, I was asked to determine the values of $\alpha$ such that for each of ...
0
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1answer
55 views

Inhomogeneous integral equation

Let $g$ be a nonnegative Borel-measurable function, that is locally integrable on $[0, \infty)$. Assume that $g$ satisfies for all $t \geq 0$ the inequality $g(t) \leq a + b \int^t_0 g(s) ds$, where ...
3
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0answers
32 views

Integral of a function over the Koch Curve. Is it rigourous enough?

(I want to investigate the validity of this approach, as I already know this is the correct result) I present a proof that $$\int_{K} (x+y) \ \mu(x,y)={{9+\sqrt 3} \over 18}$$ Where the region of ...
2
votes
1answer
40 views

Improper integral test

I am looking for a reference for this fact (or a proof): The Improper integral $\int_{1}^\infty f(x) dx $, where $f$ is positive and continuous, exists if $\lim_{x\to \infty}\frac{\log f(x)}{\log ...
1
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0answers
32 views

Show that this integral is finite $\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx$

Let $p > -1$ and $r \in \mathbb{N}$, show that $$\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx = \int_0^\infty x^p (\ln x)^r e^{-x} dx$$ and that this integral is finite. To ...
0
votes
3answers
51 views

How to find $\int_{0}^{x}\sqrt{t^2-2|t|+1}dt$

How to find $\int_{0}^{x}\sqrt{t^2-2|t|+1}dt$ where x>0. I tried to know how will the Wolfram treat with absolute term and how to find this integral, but I found that the Wolfram couldn't ...
1
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1answer
28 views

The geometry meaning of Riemann–Stieltjes integral [duplicate]

Maybe my question seems so strange but I want to know what is the geometry meaning of Riemann stieltjes integral ??
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4answers
97 views

Integration of $\int_1^2 x(2x-3)^4 \, dx$ by substitution

An exam question: $$\int_1^2 x(2x-3)^4 \, dx\\ U = 2x - 3$$ I have rearranged to get $dx = dU/2$. So I am now at $\int xU^4 \, dU$ I am not quite sure what to do with the $x$ as it is not cancelled ...
-7
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1answer
47 views

Solving Integral that includes radical expression 2 [duplicate]

I need to solve this integral analytically. I used many methods but I can’t solve it. Please help me. Thank you
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0answers
22 views

Hydrostatic Force integration problem

Hello I was trying some work problems in my calculus textbook and came across one like this http://imgur.com/xdOKTht,cv41pvt#0 http://imgur.com/xdOKTht,cv41pvt#1 And I was wondering why the area was ...
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2answers
49 views

Solving Integral that includes radical expression

I need to solve this integral analytically. I used many methods but I can’t solve it. Please help me. Thank you $$\int\sqrt{x^4-c}\ dx$$ http://i62.tinypic.com/15heux1.png
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0answers
19 views

Calculate the right Riemann sum to approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5]$.

I'm attempting to calculate the right Riemann sum and approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5] = [a, b]$. $$\sum_{k = 1}^{n}{f(a + k\Delta x)}\Delta ...
0
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1answer
30 views

Integrating a second order non homogeneous ODE

I took an exam and the teacher didn't solve this problem during the correction. I need to solve $$y''(x)-y(x)=\sin (e^x)$$I was able to find the solution to the homogeneous equation ...
4
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3answers
49 views

Evaluating $\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$ by substitution

$$\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$$ $u^2 = x^2 - 1$ I have worked out that $dx = du$ and that $u = x - 1$ so, $\int\frac{x^3}{u} du$ - but I'm stuck at this stage. Any ...
4
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0answers
45 views

What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve

What is $$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$ where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure. Attempt: I can evaluate the integral numerically and I have derived a method ...
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0answers
29 views

I need help on a few questions i have no idea how to answer [on hold]

If an object moves such that its velocity is given by m/s Use integration methods to find an equation giving the distance of the object at any time. AND The growth of a saw-tooth waveform flowing ...
-4
votes
2answers
41 views

Intergration the following functions with respect to $x$ [on hold]

$x^2\cos{x}$ $\ln{(x-1)}$ $(\ln{x})^2$ I know I must use the integration by parts to solve these questions but I have no idea at all how to continue.
0
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1answer
14 views

Prove that the image of a curve has zero content

Definition: A set $A \subset \mathbb{R}^2$ is said to have zero content if, for all given $\varepsilon >0$, exists a finite collection of rectangles $A_1, \dots, A_n$ such that $A \subset ...
0
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1answer
64 views

integration of $ \frac{1}{\ln(A \sqrt x + B)}$

I try to find solution for that: $$\int\frac{1}{\ln(A\sqrt x + B)}$$ where $A$ and $B$ are constants. I don't mind to make some assumptions or expend it, but it doesn't work with Taylor expansion. ...
0
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0answers
29 views

How CES function with integral becomes min function in the limit

I wonder how a CES function over a continuum of goods, $$\left(\int_1^\infty c(\theta)^\delta g(\theta) \mathrm{d}\theta\right)^\frac{1}{\delta}$$ where $c(\theta), g(\theta)>0 \forall ...
0
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1answer
14 views

How has this answer been derived? Integration problem

I just don't understand how my teacher has got from part 1). to 2). Where has the 1/5 come from? Q(t) = ∫(1/((t^2)−t−6))dt ...
-2
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1answer
28 views

Integration using Substitution [closed]

Firstly, I know that the graph of function, $f$ must cut the x-axis at least once such that the definite integral will equal to zero so I can apply Roelle's theorem somewhere. For b (i), letting $u ...
1
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1answer
16 views

How do boundaries change in this particular double integral?

Is there a simpler representation for an integral of the form $$\int_1^x \int_1^t f(u)\; du\; dt$$ analogous to $$\sum_{t=1}^x \sum_{u=1}^t f(u)=\sum_{t=1}^x (x-t+1)f(t)$$ ? It seems like there should ...
1
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1answer
34 views

Integration Properties

I have always had a mental block towards this property and would be truly grateful if someone would please help me. $$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$$ Consider $$f(x) = x, for ...
2
votes
1answer
114 views

Integration with Limits

Find $$\displaystyle \lim_{n \to \infty} \int^{1}_{0}(x^{n}+(1-x)^{n})^{\frac{1}{n}}dx$$ Now, the answer is $$\dfrac{3}{4}$$ Now, the solution was hinted like this: using the property ...
1
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0answers
21 views

An example in which the Fubini theorem is inapplicable

This is example 8.9(a) in Rudin's Real and Complex Analysis, (alternatively, exercise 10.2 in Rudin's Principles of Mathematical Analysis). Let $X$ and $Y$ be the closed unit interval $[0,1]$, let ...
1
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2answers
81 views

Proving that $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.

Problem. Show that a bounded function $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable if and only if for every $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that for any partition $ ...
0
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0answers
30 views

Area of a surface of revolution about the y-axis-

I'm trying to find the area of a surface of revolution generated by the curves $$y=x^3,\quad x=1,\quad x=2, \quad\rm{around} \quad y=-1 $$ \begin{array}{lcl} A &=& 2\pi \int_1^2 {(y + 1)\sqrt ...
2
votes
1answer
29 views

equality between variable and integral

I received the following question as part of my homework: Let $f(x)$ be a continuous function onto $[0,1]$. $f(x)\le\frac{1} {2\sqrt{x}}$ for every $0<x\le1$. Prove that x=0 is the only solution ...
3
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2answers
50 views

Find the following indefinite integral: $\int (x^2+6x+5)^{12} (x+3) \ dx$

The solution I got was $(1/13)(x^2+6x+5)^{13} + C$ I am not sure if I am correct though and would like help. Thanks!
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0answers
25 views

Gauss Chebyshev formula [closed]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral. $$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1. Also compare the result with true value, where the zeros and the ...
1
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2answers
29 views

Changing order of double integral

I have a double integral with the integral with respect to x on the inside between 0 and y^2 and the outer integral with respect to y between 0 and 1. If i change the order of the integrals what would ...
0
votes
1answer
32 views

Evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle.

I need to evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle traveled in the counterclockwise direction. I'm thinking about writing the function as a Laurent series and then ...
0
votes
1answer
25 views

Integration of step functions

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwartz inequality before, but I've hit a roadblock because I've no idea whether or not I can ...
4
votes
1answer
70 views

Solve integral with exponent

How to solve integral: $$\int^\infty_0e^{-\frac{At^2}{t+1}}~dt , \quad A>0$$
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votes
1answer
37 views

What is $\int_0^2\int_{y/2}^{(y+4)/2}y^2(2x-y)e^{(2x-y)^2}dxdy$? [closed]

What is $$\int_0^2\int_{y/2}^{(y+4)/2}y^2(2x-y)e^{(2x-y)^2}dxdy$$ if we change the region of integration to a rectangle?
2
votes
1answer
43 views

Calculate area of the region formed by $f(x)= x^3-x^2$ and x-axis

What is the area of the region formed by the graph of $f(x)=x^3-x^2$ and the $x$-axis in the interval $[0,3]$? Did I do this right? I get $$\int_0^3x^3-x^2\,dx$$ giving me the answer of $45/4 = ...
2
votes
0answers
85 views

Finding $\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$

Finding $$\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$$ This is a homework. I tried to solve it by assuming $x=u^2$ but after that the integrals become not simple, so I don't know how ...
6
votes
2answers
71 views

An integration question to be solved without using differentiation under the integral sign.

$$I(\alpha)=\int_0^1 \frac{x^\alpha-1}{\ln x}dx.$$ As the title says, if someone could solve this without using the differentiation under the integral sign technique, I would be very grateful.
1
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1answer
26 views

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)\hspace{1mm}|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$ As you can see two forms are easy. $$\iiint_E ...