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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Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
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2answers
18 views

Evaluate $\lim _{n\to \infty }\int_1^2\:\frac{x^n}{x^n+1}dx$

We have $$I_n=\int _1^2\:\frac{x^n}{x^n+1}dx$$ and we need to find $\lim _{n\to \infty }I_n$. Have any ideea how we can evaluate this limit?
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3answers
15 views

Given that: $T(x,y)=\ \int_{x-y}^{x+y} \frac{\sin(t)}{t}dt\ $, calculate: $\frac{\partial T}{\partial x}(\frac{\pi}{2}, - \frac{\pi}{2})$.

Given that: $T(x,y)=\ \int_{x-y}^{x+y} \frac{\sin(t)}{t}dt\ $, How do I calculate: $\frac{\partial T}{\partial x}(\frac{\pi}{2}, - \frac{\pi}{2})$? I seriously have no direction for how to solve ...
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2answers
39 views

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$ Can I multiply and distribute the $ \ x^{7/2}\ $ and $ \ sec^2 \ $ together. What is the strategy to solve this problem.
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1answer
14 views

Initial value problem through origin

$\frac{dz}{dt}=8t*e^z$, Through the origin I have never done an initial value problem before, but I took it to mean that it gave me the initial value of the differential equation (0, 0) and that I ...
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0answers
15 views

Stieltjes Integral - If $f, f^2, g, g^2\in R(\alpha)$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$

My question is if $f, f^2, g, g^2\in R(\alpha)$ on $[a,b]$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$ as well? This question stemmed from a problem in Apostol's Analysis, in ...
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1answer
28 views

How can I prove fundamental theorem in measure theory

How can I prove that $1)$ if $f$ and $g$ are measurable functions such that $0 \le f \le g$ Then $\int_{X}{}fdμ \le \int_{X}{}gdμ$ $2)$ if $f$ and $g$ are integrable functions such that $f \le g$ ...
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2answers
22 views

Find solution to the differential equation

$\frac{dB}{dx}+2B=50$ $B(1) = 50$ I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.
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3answers
38 views

Integral of $\cosh^3(x)$

What is the integral of $\cosh^3(x)$? And how exactly can I calculate it? I've tried setting $\cosh^3(x)=(\frac{e^x+e^{-x}}{2})^3$ but all I get in the end is one long fraction.
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1answer
11 views

Showing something involving integrals is an inner product

I have this problem: Let $C([0,1])$ be the real vector space of continuous functions on the interval [0,1]. Show that $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
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1answer
38 views

Integrate $\int_{0}^1 (1 + 4y^2)^{1/2} dy$ [duplicate]

$$\int_{0}^1 (1 + 4y^2)^{1/2} dy$$ So, how do I integrate this without the use of trigonometrical substitution? Can anybody give me a hint? Thank you!
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1answer
58 views

Trouble solving an integral

So I have been trying to solve this equation, The given answer is, I began by using substitution to change the integral. Substituting t back in where t is taken from 0 to infinity. ...
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1answer
96 views

Poisson Integral relation

If $$ I_n(r) = \int_0^\pi \frac{\cos nx}{r^2-2r\cos x+1} \, dx $$ How to prove that $$ I_{n-1}(r)+I_{n+1}(r)= \left(r+\frac{1}{r}\right)I_n(r)\text{ ?}$$ I only find that $$I_{n-1}(r)+I_{n+1}(r)= ...
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0answers
17 views

Quality of approximation of an Ito integral

How could I investigate whether $$P(t,T-t)\left[a(T-t-\Delta)-a(T-t)+ (b(T-t-\Delta)-b(T-t))'x(t)+ \frac{1}{2}b(T-t)'\sigma\sigma'b(T-t)+ b(T-t)'(x(t+\Delta)-x(t))\right]$$ is a good or bad ...
2
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1answer
35 views

A question on integration of differential forms on a manifold

I'm fairly new to differential geometry and have been reading up on integration on manifolds. All the texts/lecture notes that I've read so far always consider integrating an $n$-form over an ...
2
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2answers
32 views

Need help solving complicated integral $\oint_{\mathcal C}\begin{pmatrix}x_2^2 \cos x_1 \\ 2x_2(1+\sin x_1)\end{pmatrix} dx$

Let $\mathcal C$ be the curve that traces the unit circle once (counterclockwise) in $\mathbb R^2$. The starting- and endpoint is (1,0). I need to figure out a parameterization for $\mathcal C$ and ...
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1answer
35 views

Understanding a particular transformation of an integral given in a proof

Using the theorem of mean values find the sign of the integral... $$\int_{0}^{2 \pi}{\sin x \over x}dx= \int_{0}^{\pi}{\sin x \over x}dx+\int_{\pi}^{2 \pi}{\sin x \over x}dx$$ Then: $[x-\pi=t ; ...
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3answers
39 views

Evaluate the integral $\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x$

Evaluate the integral $$\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x.$$ If I transfer $(x^2 + 4)^5$ to the numerator, how do I integrate?
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2answers
57 views

Show that the antiderivative exist [on hold]

I am new to this. How do I show that the antiderivative exist and show that is continuous too? Thanks
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2answers
43 views

Evaluating a limit expression

I am trying to show the following identity but I am stuck. $$\lim_{t\nearrow 1}(1-t)\int_0^t\frac{g(s)}{(1-s)^2}\,ds = g(1)$$ for any $g \in C[0,1]$. Apparently, the proof follows from L'Hopital's ...
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1answer
12 views

Finding marginal distribution integration help

Let: $f_Y(y)=e^{-y}$ Let: $ \mathbf P(X=k$ | $Y=y)$ = $\binom{2}{k}(e^{-y})^{k}(1-e^{-y})^{2-k}$ where k = 0, 1, 2 To find the density of $X$: $f_X(k) = \int_0^ \infty ...
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1answer
51 views

Explanation for absolute value

So $f_a:R\rightarrow \:R,\:f_a(x)=\:\frac{1}{\left|x-a\right|+3}$, and we have to evaluate $\lim _{a\to \infty }\int _0^3\:f_a\left(x\right)dx$. But $\left|x-a\right|\:$ is equal with: ...
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1answer
22 views

Riemann integral property proof using the definition

We say that a function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if for every $\epsilon>0$, there are two step functions $g_1,g_2$ such that $g_1 \leq f \leq g_2$ and $\int_a^b ...
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2answers
47 views

Find the solution to the differential equation

Assume $x\gt 0$ $$x(x+1)\frac{du}{dx} = u^2$$ $$u(1) = 4$$ I started off by doing some algebra to get: $$\frac{1}{u^2}du = \frac{1}{x^2+x}dx$$ I then took the partial fraction of the right side of ...
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0answers
9 views

Under what assumptions is the following first moment monotone?

I'm working on an economic model and am encountering the following mathematical issue. Let $x\sim \mathcal{N}(\mu,1)$, and define $$V(\mu)=\int_0^{\hat x(\mu)}x ...
0
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1answer
16 views

Extending the definition of curve length

I know for continuously differentiable curves on closed interval $[a,b]$, the curve length is given by $\Lambda (\gamma)=\int_a^b |\gamma^{'}(t)|dt$. But what about curves such that $\gamma^{'}(t)$ is ...
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7 views

Substitution in complex-valued Fourier integral

In Knapp (Representation theory of semisimple groups, 86'), on page 34 it is shown by means of Euclidean Fourier transform that the principal series representation of $SL(2, \mathbb C)$ is irreducible ...
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0answers
51 views

Find the upper and lower sum of an integral with a floor

I'm having some trouble and looking for some help with a problem i'm trying to solve. Without the floor function it would be easy but the floor has made it a bit trickier: Find the upper and lower ...
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0answers
37 views

Integrating a rational function of exponentials

Let $\gamma ,\mu > 0$ be positive real constants and $\beta \in \mathbb{R}$ be a real constant. How can I evaluate the following indefinite integral? $$ \int \frac{e^{2\gamma t} (e^{-\mu t} - ...
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0answers
32 views

Is Randall's model for a sling correct? [on hold]

In What If 116 Randall Munroe talks about ways to get drivers around a race track The 13th paragraph he imagines the drivers at the end of what seems like a big tether ball: Imagine a "vehicle" ...
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1answer
79 views

Integrate 1/ln(ln(x)) asymptotically

I was looking for the asymptotic behaviour of the anti-derivative of $\frac{1}{\ln \ln x}$, in terms of the big-O notation. Wikipedia's list does not have this integral, and Wolfram Alpha says "no ...
2
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1answer
31 views

Fourier Transform of $\delta(t-nt)$

Given the discrete signal $x(n)=\begin{bmatrix} \alpha ^n, n\geq 0 \\0, n<0 \end{bmatrix}$ where $\alpha \in (-1,1)$ and some natural number $N$, we know that the discrete signal $y(n)$ (where $0 ...
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1answer
22 views

Let $f\colon [a,b]\to\mathbb R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$.

Let $f\colon[a,b]\to \mathbb R$ is continuous and $$G(x,t)=\begin{cases}t(x-1)&\text{when $t\leq x$,}\\x(t-1)&\text{when $t\geq x$.}\end{cases}$$ Let $$g(x)=\int_0^1f(t)G(x,t)\,\mathrm dt.$$ ...
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37 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
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0answers
86 views

How to compute this triple integral?

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
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0answers
31 views

Integral of $\frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}$

Does the following integral have a closed form solution? $$ \int_{0}^{\infty} \frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}{\rm d}x $$ where $\alpha$, $\beta$ and $x_0$ are ...
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1answer
22 views

Why is just $0$ extreme point? v22

We have $f:R\rightarrow R,\:f\left(x\right)=x^3-3x+2$ and we need to find extreme points for $g:R\rightarrow R\:,\:g\left(x\right)=\int _0^{x^2}\:f\left(t\right)e^tdt$. Here is all my steps: ...
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6answers
71 views

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically? Using integration by parts I got the form: ...
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1answer
72 views

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ [on hold]

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ and negative for $|x|<1$ ? I don't understand .. I can't see.. damn it!
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0answers
16 views

Compute the integral $y^2dx -2xydy$ with the curve of an equiliateral triangle with vertices at $(0,0), (2,0)$ and $(1,-2)$

The partial derivates are equal so I thought to use the theroem of line integrals but I also thought using Green's Theorem would be easier or possibly evaluating the each segment of the traingle ...
2
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1answer
27 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
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0answers
20 views

Computing the value of a line integral of a vector field in the plane

We are given the vector field $ x^2dx+y^2dy $ and are interested in the line integral of it over the closed equilateral triangle with vertices (0,0) (2,0) (1,-2) Because the partial derivatives of ...
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3answers
126 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
2
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1answer
26 views

How can I find monotonicity intervals? v18

We have $F:\mathbb{R}\rightarrow \mathbb{R}$, $F(x)=x\int _0^x (1+\cos(t)) \, dt$ and we neeed to find monotonicity intervals and I don't know how... Here is what I try to do: $$F'(x)=\int _0^x ...
2
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1answer
40 views

Where do the step function integral boundaries come from?

EDIT: I have a confusion about Heavyside step function. Suppose I have integral like $$ \int_{0}^{\infty}dE_1\int_{0}^{\infty}dE_2\int_{0}^{\infty}dE_3 \delta(2- \gamma-E_1-E_2-E_3) $$ my first ...
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2answers
21 views

How we can prove that $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n f(t)\:dt$ is convergent?

We have $f:\left(-1,\infty \right)\:\rightarrow \:R,\:f\left(x\right)=\frac{x}{x+1}$ and we need to prove that: $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n\:f\left(x\right)dx$ is convergent.Maybe, in ...
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2answers
64 views

Can somebody please show me the necessary steps to solve this Calculus problem?

I have a homework assignment that asks me to solve the differential equations and it gives me: \begin{align*} xy^2y' & = 2-x\\ y''+4y & = 8x\\ y(1)& =1 \end{align*} Are these three ...
3
votes
3answers
69 views

Evaluation of $ \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos xdx $

Evaluate $$\displaystyle \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos x \,dx .$$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int \left(\cos 2x \right)^{\frac{11}{2}}\cdot ...
0
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2answers
43 views

Why Riemann sum is convergent? [on hold]

Why $\frac{1}{n}\sum _{k=1}^nf\left(\frac{k}{n}\right)$ is convergent? I don't understand how we can prove that is bounded and monotone... For instance: $f:R\rightarrow R,\:\:f=\frac{1+x}{1+x^2}$, ...
-6
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2answers
44 views

Two indefinite integral problems. [on hold]

Please help me out in these $$\int \frac{dx}{1-3\sin (x)}.$$ Second $$\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx.$$