All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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29 views

integral alternative of $\sum f(p)$ from $(1<\text{all primes}\leq n)$ to $(\text{maximum prime}<n)$

I have a naive question that if someone could find the integral alternative of $$\sum_{\substack{2\le p\le n\\p\text{ is prime}}} f(p)$$ where f(p) is a non-decreasing monotonic function.
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1answer
50 views

How do you find the distribution of this sum?

If $X\sim \text{Normal}(\mu=0, \sigma^2), Y\sim \text{Unif}(0,\pi)$, and $X \perp Y$, how do you find the distribution of $Z=X+a\cdot cos(Y)$ for some $a > 0$ ? I've found the distribution ...
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0answers
27 views

Is there a way to find limits of integration numerically for triple integrals?

I can do double integrals changing order of integration very easily because the drawing of graph is fairly easy in 2 dimensions and I use the graph to help me with finding limits of integration for ...
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1answer
35 views

Volume of $M = \{(x, y, z) \in \mathbb R^3 : z^2 \le x^2 + y^2 \le r^2, z \ge 0\}$

I am having a hard time solving this. Is $M$ 2D or 3D? Isn't it an infinite cylinder which starts at the $x,y$ axis and grows vertically with $z$? Or is it just a circular disk at $x,y$ axis? We are ...
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3answers
128 views

Theorem 6.20 rudin Integration

How does he do the algebra? (page 134 Rudin, chapter 6 ,theorem 6.20) $\left| \frac {F(t)- F(s)}{t-s} -f(x_o) \right| = \left| \frac{1}{t-s} \int_s^t[f(u) - f(x_o)]du \right|< \epsilon $ also, ...
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2answers
62 views

Integral Simplification

I was hoping to get some help.... I have a complex integral expression: $\frac{\int_0^\infty t\left( A(0)\alpha \left( b+1 \right) {e}^{-\alpha t \left( b+1\right)} + A(0) \frac{b}{g - 1- b } \left( ...
1
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1answer
110 views

Problem involving lower and upper sums in intergration

If $f$ is integrable on $[a,b]$, prove that for any partition $P$, $LS(f,P)\leq \int_{a}^{b}f \leq US(f,P)$. My attempt: Since $f$ is integrable on $[a.b]$, by definition there is some number ...
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1answer
101 views

Is it possible to solve problem 4 (double integral) without using a graphing calculator?

I know how to solve problem 4 but I cheated by using the Wolfram Alpha graphing calculator. Is there any way to solve it without a calculator?
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1answer
78 views

The slope of the tangent line

I don't know how to solve these three, especially the first and the second one: $1.$ $x=0$ means upper limit also equal to $0$? what should I do to deal with the $(t^2+\pi^2)^{1/2}$? $2.$ Find the ...
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3answers
106 views

separation of variables: integral of 1/quadratic

I have a differential equation $${\frac{dx}{dt}}=x[a-(b+c)]-ax^2$$ where a,b and c are positive constants and there is initial condition: $$x(0)=x_0$$ which I need to solve by separation of variables. ...
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0answers
164 views

Absolutely Integrable

I don't know if this is a silly question, but if I have $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and I know that $\int_{\mathbb{R}^n}|f|$ exists, can I say that $\int_{\mathbb{R}^n}f$ also exists? ...
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5answers
151 views

$\int_{0}^{\pi} \exp\left(\cos\left(t\right)\right)\cos\left(\sin\left(t\right)\right)\,{\rm d}t=\pi$

Does anyone have a proof of the above integral? I have one proof, but I wanted to see other proofs.
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7answers
347 views

integral of $x^2e^{-x^2}~dx$ from $-\infty$ to $+\infty$

I know that the $$\int^{+\infty}_{-\infty}e^{-x^2}~dx$$ is equal to $\sqrt\pi$ It's also very clear that $$\int^{+\infty}_{-\infty}xe^{-x^2}~dx$$ is equal to 0; However, I cannot manage to ...
1
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1answer
76 views

How to integrate this? Im stuck. Thanks :)

$$\int\dfrac{t-2}{t+2-3\sqrt{2t-4}}dt$$ I don't know how to do this question. Can anyone help me to solve this? I'm not sure whether want to use substitution or what.
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0answers
52 views

Determining two sets of boundary conditions for a double integral problem in the polar coordinate system

There are two sets of boundary conditions that you can use to solve this problem in the polar coordinate system. Set 1: $$\theta=0\quad to\quad \theta=\frac{\pi}{2}\\r=0\quad to\quad ...
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1answer
117 views

Calculating volume bounded by area

The problem is calculation of the volume bounded by $(x^2+y^2+z^2)^2=\frac{64}{x^2+y^2}$ I tried using cylindrical coordinates: $x=r*cos(t) ; y=r*sin(t); z=z$ then this gives me: ...
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2answers
535 views

Evaluating the integral $\int\frac{2x^3+x^2+1}{x^2+x-2}\space dx$

As suggested in the title, I am given the following integral: $$\int\frac{2x^3+x^2+1}{x^2+x-2}\space dx$$ I have tried to solve it several times, but I wind up with the wrong answer, although its ...
1
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1answer
26 views

i want prove when $f$ and $f_{n}$ are non-negative then $\int f\quad d\mu=\sup_{n} \int f_{n}\quad d\mu=\lim_{n\to \infty}\int f_{n}\quad d\mu$

let $f$ and $f_{n}$ are non-negative extended real-valued measurable functions and $f_{n}(\omega)\nearrow f(\omega)$, $\forall \omega \in \Omega $ ,then $$\int f\quad d\mu=\sup_{n} \int f_{n}\quad ...
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2answers
28 views

What does d(0, P) notation mean?

In the solution of this example, the author uses the following: $$d(0, P)=|r|$$ But what does that notation mean?
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1answer
258 views

Integrals involving Hermite Polynomials

Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial ...
3
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1answer
59 views

Improper Integral in an old manuscript

I was reading through an old manuscript and came across the following "elementary exercise:" $\int^\infty_0 \dfrac{1}{1+x^2 \sin^2 x} dx$. Anyone have a clever way of seeing this? I haven't done ...
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1answer
29 views

Showing a set is orthonormal using an integral

Let $V$ (which is infinitely dimensional) be the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$. Show that $V$ is a vector space. Define $\langle-,-\rangle: V\times V\to \Bbb{R}$ by $$\langle ...
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4answers
92 views

Equal integral but only one of them converges absolutely .

Consider the following integral $$\int_0 ^\infty \frac{\sin x}{1+x} \, dx.$$ By integration by parts we get $$\int_0^\infty \frac{\cos x}{(1+x)^2}\,dx.$$ But according to Rudin , one of them is ...
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0answers
36 views

Parseval's Theorem to estimate number of terms

Question: Find the Fourier series for $f (x) = x^2$ on the range $(−\pi, \pi)$. Use Parseval’s theorem to estimate approximately how many terms of the Fourier series are required to be ...
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1answer
43 views

Integration involving non-elementary functions

Let \begin{equation}g(t)=\begin{cases}\frac{\sin{\frac{1}{2}}t}{t}, & t \not =0 \\ \frac{1}{2}, &t=0 \end{cases} \end{equation} Calculate $\text{lim}_{m\to ...
1
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0answers
32 views

Integral coordinate change

I'm trying to compute $$\int_0^n\int_0^n\int_0^n\frac{\sin(x-y)\sin(y-z)\sin(x-z)}{(x-y)(x-z)(y-z)}\,dx \,dy \,dz$$ I want to make new coordinates as $a=x-y$, $b= y-z$ (so $x-z=a+b$) and $c=z$. I'm ...
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0answers
44 views

Non-negative, continuous function with integral [duplicate]

Let there be an integrable, non-negative function $f$ in a range $[a,b]$. If the integral $\int_a^b f(x) \, dx$ equals $0$, prove that $f(x)=0$ for every $x$ for which $f$ is continuous. I have ...
3
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2answers
101 views

Riemann integration of an odd function

$f(x) = {\rm sgn}({\rm sin}(\frac{\pi}{x}))$ if $x \neq 0$ and $f(0) = 0$ where ${\rm sgn}(x) = 1$ if $x > 0$, ${\rm sgn}(x) = −1$ if $x < 0$ and ${\rm sgn}(0) = 0$. Show $f$ is Riemann ...
3
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2answers
258 views

How prove this integral $\int_{0}^{\pi}\dfrac{1}{\cos{\theta}-\cos{x}}dx=0$

Question: show that $$I=\int_{0}^{\pi}\dfrac{1}{\cos{\theta}-\cos{x}}dx=0$$ where $0< \theta<\pi$ My try: since $$\cos{\theta}-\cos{x}=2\sin{\dfrac{\theta-x}{2}}\sin{\dfrac{\theta+x}{2}}$$ ...
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1answer
97 views

Show $\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0$

Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that ...
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1answer
66 views

Derivative of step function is Dirac delta function

The derivative of a locally integrable function $f$ is by definition the linear functional $$\phi \mapsto -\int_\mathbb{R} \phi'(x)f(x) dx$$ Using this definition, why is it that the derivative of ...
4
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1answer
217 views

How to integrate $e^{\sin x}(x \cos x - \tan x \sec x)$

How would on find the indefinite integral $e^{\sin x}(x \cos x - \tan x \sec x)$ Our professor gave it to us as a review question. He told us it was from an exam several years ago as extra credit, ...
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0answers
53 views

Is it always possible to find a $t>0$, such that $\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|dx<C~~~?$

Is it always possible to find a $t>0$, such that $$\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|\,dx<C~~~?$$ where $C$ is independent of $n$. Here is my idea: We know that \begin{align} ...
0
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1answer
51 views

Compute for Cov(X,Y) and Correlation(X,Y)

Let $(X, Y)$ be uniform on the half disc $D = \{(x, y) : 0 < y, x2 + y2 < 1\}$. How should I approach this problem. Should I solve double integral with inside goes from $-\sqrt1-x^2$ to ...
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2answers
75 views

Indefinite Integrals?

Need help finding the anti derivatives of $\dfrac{\sqrt{x}}{3}$ and $\dfrac{-3}{5x}$. I don't know what to do when the problem is in a fraction.
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1answer
65 views

system of differential equations for spread of infection query

I have been given a set of equations describing the spread of infection in a population: $${\frac{dS}{dt}}=-aIS+bI+c-cS$$ $${\frac{dI}{dt}}=aIS-bI-cI$$ where S and I are susceptible and infected ...
0
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1answer
62 views

Exercise on Riemann-Stieltjes integral

I am trying to solve the following exercise: Let $f:\mathbb R \to \mathbb R$ be a continuous function such that for every $a<b$, the integral $\int_a^b fdf$ exists and is $0$. Prove that $f$ is a ...
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1answer
108 views

Evaluate the indefinite integral!

$\displaystyle \int \frac{\tan(\ln(x^5))}{x}\, dx$. Please help me evaluate this! I'm having trouble figuring out what "u" should be using "u" substitution.
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1answer
51 views

Supremum of Expectation of a Sequence of Random Variables

Let $(X_1, X_2,...)$ be a sequence of random variables that converges almost surely to a random variable $X$. Show that if $\sup_n EX_n^2 < \infty $, then $EX^2 < \infty$. I believe this is ...
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3answers
97 views

Evaluate the indefinite integral

Evaluate the indefinite integral. I am having trouble. $$\int\frac{dx}{x\ln\left(7x\right)}$$ Help me. Please!
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1answer
45 views

Which of the following double integrals would correctly solve this problem?

I obtained two sets of boundary conditions. Set 1: $$x=-\sqrt{4-y^2}\quad (for\quad x<0)\quad to\quad x=\sqrt{4-y^2}\quad (for\quad x>0)\\y=-2\quad to\quad y=2$$ Set 2: $$x=-2\quad to\quad ...
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3answers
103 views

Without performing the integral to show

Without performing the integral, show that $$ \int_{-2}^{2}e^{x^2}x^3dx=0 $$ Please help me. I do not know where to start.
3
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1answer
78 views

How does one prove this formula?

Show that $$\int_0^{\infty}\frac{1}{\sqrt{x}}\exp\left[-a^2 x\left(\frac{x-6}{x-2}\right)^2\right]\,dx=\frac{\sqrt{\pi}}{a},$$ where $a$ is real.
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1answer
22 views

Alternate boundary conditions for this double integral?

(Ignore the writing above Example 1; this question is based on Example 1 only) I am trying to practice my ability to produce boundary conditions. How are these boundaries below? $$x=1\quad ...
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5answers
351 views

Integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}}{1+\cos^4{x}}dx$

Question: Find the integral $$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ my try: since $$I=2\int_{0}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ then I can't. I ...
2
votes
1answer
47 views

What am I doing wrong in this attempt to solve this integral? [duplicate]

The correct solution: My attempt: $$\int_{\sqrt{y}}^{1}y\ln x \, dx\\=y\int_{\sqrt{y}}^{1}\ln x \, dx\\=y\cdot\left.\frac{1}{x}\right|_{\sqrt{y}}^{1}\\=y-\sqrt{y}$$
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2answers
55 views

Can someone explain how integration by parts was done?

I just would like someone to identiy u and dv.
0
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0answers
54 views

Changing the boundaries in this double integral problem.

I am going to attempt this but with the following modification to the boundaries: $$x=0\quad to\quad x=sin^{-1}y\\ y=0\quad to\quad ...
1
vote
2answers
87 views

Why can't I change order of integration when computing double integral?

$$m=\int_{sinx}^{cosx}\int_0^{\pi/4}ydxdy\\=\int_{sinx}^{cosx}\left.yx\right|_{x=0}^{x=\pi/4}dy\\=\int_{sinx}^{cosx}\frac{\pi y}{4}dy\\=\frac{\pi(cos^2x-sin^2x)}{8}$$ Shouldn't you get the same ...
0
votes
1answer
135 views

Surface Area of a Parametric Curve

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. $$ ...