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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0
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2answers
12 views

Integral of $\frac{e^x}{5+2e^x}$

Regarding the integral of this term$\frac{e^x}{5+2e^x}=\frac{e^x}{2(\frac{5}{2}+e^x)}$ Is the answer $\frac{1}{2} \ln(\frac{5}{2} +e^x)$ or $\frac{1}{2} \ln(5+2e^x)$? When I substitute $u= ...
0
votes
1answer
14 views

Using Taylor series find derivatives of arctan(x)

Using Taylor series for $arctan(x)$, find $f^{(5)}(0)$ and $f^{(6)}(0)$ for $f(x)=arctan(x)$ I figure for this problem I compare the general Taylor series formula to the Taylor series for $arctan(x)$ ...
0
votes
0answers
20 views

How do i evaluate the following integral?

Hi I was wondering if someone can help me evaluate the following integral. Show that if $-1 < x < 1$, then $$\int_{0}^{\pi} \frac{\log{(1+x\cos{y})}}{\cos{y}}dy= \pi \arcsin{x} $$ thank you ...
3
votes
1answer
19 views

Find $\lim_{n \to \infty} n^{\alpha} \int_{n}^{\infty} \frac{f(x/n^2)}{x^{\alpha + 1}}(x-n)dx$

I am looking at an old exam in my measure theory and integration class. I am trying to solve a problem and am wondering if I am doing it right. Problem Let $f$ be a bounded measurable function on ...
-1
votes
5answers
107 views

how does one integrate: $\int \frac{1}{x^2-a^2}dx$

how does one integrate: $$\int \frac{1}{x^2-a^2}dx$$ I know it looks very similar to the known formula $$\int \frac{1}{x^2+a^2}dx$$ but it doesn't help really. note: Im not allowed to use the ...
0
votes
3answers
23 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 ...
0
votes
0answers
31 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
1
vote
3answers
59 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?
0
votes
1answer
37 views

How do we prove $\int \frac{\ln(1+x)}{x}dx = -\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$?

After working on the integral $\int_{0}^{1} \frac{\ln(1+x)}{x}dx$ for a couple of hours, I became convinced its antiderivative was not elementary. So I looked it up on Wolfram Alpha, and it found that ...
3
votes
2answers
54 views

Hint on how to find $\int \frac{x^2}{1+x^2}dx$

I am almost sure that this would have been asked before, but how can one find $$ \int \frac{x^2}{1+x^2} dx? $$ If I had a $x^2 - 1$ in the denominator, then I could factor into $(x-1)(x+1)$ and use ...
2
votes
1answer
37 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 $f \le g$ ...
2
votes
0answers
18 views

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 ...
3
votes
0answers
87 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 ...
1
vote
2answers
44 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.
2
votes
0answers
53 views

Value of improper Integral

I need help in finding the value of the integral $$\displaystyle \int_0^\infty \left(\frac{x^2}{1+x}\right)^{n-1}e^{-tx}dx,$$ where $n$ is a positive integer and $t$ is a positive real number.
1
vote
1answer
15 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 ...
3
votes
0answers
17 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 ...
2
votes
1answer
59 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. ...
1
vote
2answers
24 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.
0
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3answers
41 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.
2
votes
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 ...
1
vote
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
1
vote
1answer
13 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 $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
2
votes
1answer
43 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!
7
votes
1answer
121 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)= ...
1
vote
1answer
36 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 ; ...
3
votes
1answer
70 views

Trouble solving $\int\frac{x}{\sqrt{x^2-6x}}$

I need to solve the following integral $$\int\frac{x}{\sqrt{x^2-6x}}$$ I started by completing the square, $$x^2-6x=(x-3)^2-9$$ Then I defined the substitution variables.. ...
7
votes
1answer
157 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
3
votes
0answers
521 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
1
vote
1answer
23 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 $$g(x)=\int_0^1f(t)G(x,t)\,\mathrm dt.$$ ...
6
votes
1answer
95 views

Find the closed form of the digamma related series

The question I asked here Computing $\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$ made me think to ask for your support for ...
0
votes
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 ...
1
vote
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 ...
2
votes
2answers
33 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 ...
1
vote
3answers
42 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?
6
votes
4answers
419 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi = \int_{0}^{1}\frac{x^{4}\left(1 - x\right)^{4}}{1 + x^{2}}\,{\rm d}x $$
1
vote
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: ...
2
votes
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 ...
0
votes
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 ...
4
votes
2answers
48 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 ...
6
votes
3answers
92 views

How to integrate $(e^x + 2x)^2$?

I need to integrate $\int(e^x+2x)^2dx.$ I tried breaking it into $\int(e^x+2x)(e^x+2x)dx$ and then integrating by parts, but got stuck at $$ \int (e^x + 2x)^2\,dx = ...
0
votes
1answer
25 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 ...
0
votes
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
votes
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 ...
0
votes
0answers
52 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 ...
16
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
63 views

On definition of Riemann integral

Let $I = [a, b]$ be the finite closed interval of $\mathbb R$. A partion $P$ of $I$ is a finite sequence $a = a_0 \lt a_1 \lt ... \lt a_n = b$. We write $P\le Q$ if $P \subset Q$ where $P, Q$ are ...
1
vote
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} - ...
0
votes
0answers
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 $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$