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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3
votes
1answer
70 views

Show that $\int h_n'(x) \varphi(x)\, dx \to \langle \delta, \varphi\rangle$ - Generalized functions theory

In the book Partial Differential Equations by Robert Strichartz, there's an exercise (#$1$, page $9$) that I am not quite sure how to solve. Is there anyone could give me the principal steps how to ...
3
votes
5answers
153 views

Evaluating the definite integral $\int_0^3 \sqrt{9- x^2} \, dx$

I have been having a problem with the following definite integral: $$\int_0^3 \sqrt{9- x^2} \, dx $$ I am only familiar with u-substitution and am positive that it can be done with only that. Any ...
-1
votes
2answers
65 views

show that $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
0
votes
1answer
44 views

Prove $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
8
votes
4answers
167 views

How does one integrate $x^2 \frac{e^x}{(e^x+1)^2}$?

How can I show this? $$ \int_{-\infty}^{\infty} x^2 \frac{e^x}{(e^x+1)^2} dx = \pi^2/3$$ I tried applying residuals, but the pole is of infinite(?) order.
0
votes
1answer
21 views

Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
-1
votes
1answer
31 views

How to solve simultaneous inequalities (reasked)? [duplicate]

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1≤2x+y≤2$$ $$0≤x−2y≤1$$ ...
0
votes
2answers
44 views

Domain of Integral $\int_{5}^{x} \frac {dt}{(1-t^2)}$

A function reads $$ F(x) = \int_{5}^{x} \frac {dt}{(1-t^2)} $$ Barrons says that the domain of F must be that $x >1$. But why can't $x$ be less than $1$ as well? As long as $x$ does not equal ...
2
votes
2answers
94 views

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.
0
votes
1answer
31 views

Enlargement of area and perimeter in a rotation body

Let $f: [0,1] \to \mathbb{R}$ a continuous, differentiable function with $f \ge 0$. Rotate the graph of $f$ around the x-axis. Define this rotation body in $\mathbb{R}^3$ with $A$ and the area in ...
0
votes
0answers
31 views

Need help integrating $\frac{1}{160}\log \left(5x-25\right)\left(\left(y-146\right)^2+\left(x-7\right)^2\right)=10$

The project is quite simple that I am making much more difficult because it is fun to make things difficult. Create a single function for a water tower design with a narrow part and reservoir at least ...
-1
votes
0answers
17 views

Integral becoming anti derivative [on hold]

How does integral become anti derivative? I know the proof of it.I want to make the concept clear through graph or some nice simple sentences
0
votes
1answer
21 views

Help Computing integral of continuous function

Consider the Dirichlet function $F:R \rightarrow R$ given by $f(x):=\begin{cases} x &\text{if } x < 1 , \\{}\\ x+1 &\text{if } 1<= x <= 2, \\{}\\ -x+5 &\text{if } 2<x ...
0
votes
0answers
27 views

problem solving these differential equations

$$(\sin^2(x) D^2 +\sin(2x) D+\cos^2 x+1)y=\sin^3 x$$ and there is another $$(xD^2-x(x+2)D+(x+2))y=x^3-2x+1$$
2
votes
1answer
130 views

Why does the hard-looking double integral $\int_{0}^{1}\int_{0}^{1}\frac{x(1-x)y(1+y)}{(1-xy)\ln(xy)}dxdy=-\frac{1}{2}$?

(1) originate from here problem 11322 (1) $$\int_{0}^{1}\int_{0}^{1}\frac{x(1-x)y(1+y)}{(1-xy)\ln(xy)}dxdy=-\frac{1}{2}$$ On my recent post see here Marco Cantarini and hints from other proved the ...
-1
votes
1answer
30 views

Integral convergence $\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}}$

I've tried to partial this integral from 0 to 1, 1 to e, and e to infinity. $$\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}} dx$$
0
votes
1answer
29 views

Why is this substitution not valid?

I can't seem to notice the mistake in this calculation: $$\int(5+\cos t)\sin t\:dt=-\int(5+\cos t)\:d(5+\cos t)=-\frac{1}{2}(5+\cos t)^2+k$$ I would argue I did nothing illegal because $$d(5+\cos ...
-1
votes
1answer
31 views

Find the derivative and integral of the following function

I'm a bit confused on how to work out this question, so if you could show working it would be much appreciated. Thanks. Find $f'(x)$ and $\int f(x)\,dx$ for $$f(x)= ...
0
votes
1answer
27 views

Convolution of Gaussian and error function

I am trying to evaluate the following integral: $$ \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx $$ where $$ \Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right) $$ I have ...
1
vote
1answer
27 views

How do I represent $f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$ in an integrated form?

Given the following function, how do I define it without the integral symbol? $$f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$$ I don't understand how I determine when $x + t$ is positive and ...
1
vote
1answer
59 views

Prove $\int_{0}^{1}\int_{0}^{1}[-\ln(xy)]^s\left(\frac{1}{\ln(xy)}+\frac{1}{1-xy}\right)dxdy=\Gamma(s+2)\left[\zeta(s+2)-\frac{1}{s+1}\right]$

I got the idea from here $(1)$ yield the same result as Hadjicostas. Is this $(1)$ same as Hadjicostas but just write in a different way? ...
1
vote
1answer
39 views

Evaluating the arc length integral $\int\sqrt{1+\frac{x^4-8x^2+16}{16x^2}} dx$

Find length of the arc from $2$ to $8$ of $$y = \frac18(x^2-8 \ln x)$$ First I find the derivative, which is equal to $$\frac{x^2-4}{4x} .$$ Plug it into the arc length formula ...
1
vote
1answer
24 views

How to find the limits or boundary of integration between two points?

Im trying to calculate the work done of a field between two points. The thing im struggling with is find the limits of integration. ie. $$\int_a^b$$ Can anyone help, if the points were A(1,2) and ...
0
votes
1answer
58 views

Integral of $\frac{1}{(x^2+2)^3}$

Ive been struggling to find the integral $\frac{1}{(x^2+2)^3}$ by using the integral $I_n=\frac{1}{(x^2+1)^n}$. (assume I know how to solve $I_n$ by a recursive way. Ive tried to make it to the form ...
1
vote
0answers
27 views

Differentiation of an integral in regards to different variables

It is known by the second fundamental that $$\frac{d}{dx}\int_0^x{\sin{(a \cdot t)}\ dt}=\sin{(a \cdot x)}$$ But what can we say about $$\frac{d}{da}\int_0^x{\sin{(a \cdot t)}\ dt}=\ ?$$
0
votes
1answer
23 views

Taylor series integration

I am having trouble with the following question: Integrate the Taylor series $$e^{(-t^2)} = \sum^\infty_{n=0} \frac{(-t^2)^n}{n!}$$ term-by-term to obtain the Taylor series for erf (error function) ...
0
votes
2answers
44 views

Complex logarithms when computing real-valued integral

My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral \begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx \end{equation} ...
7
votes
1answer
66 views

$ \int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx$

What is the value of $$ \int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx $$ I have no idea how to start with this integral, any hint?
0
votes
1answer
33 views

More double integration help.

Im having trouble integrating this function. $$\int_0^{0.5}\int_0^{\pi/4} \frac{r\sin\theta \ln(1-r\cos\theta -r\sin\theta )}{r\cos\theta-\sqrt{(r\cos\theta)^2+(r\sin\theta)^2 }} r \,d\theta \, dr$$ ...
0
votes
0answers
18 views

Show that a step function doesn't exist such that …

I am stuck in the 3rd question of this exercise: Let f be a function: $ f:[0,1] \to IR $ defined as : $f(2^{-m}) = 1$, $ f(x) = 0, x\not= 2^ {-m}$ ,$m \ge 1 $. 1) For n > 1: $$ S_{n} = \{ {0, ...
0
votes
1answer
13 views

Evaluating integral of product of two periodic functions

Is there a straight-forward way to evaluate an integral of the following type? $\int_0^T e^{ik\frac{2\pi}{T} t} f(t) dt, \quad k \in \mathbb{Z}$ Here $f(t)$is also a periodic function of period $T$ ...
0
votes
1answer
21 views

Double integral over a parallelogram (Fubini's theorem application)

Using Fubini's theorem in $\Bbb R^2$ and, I guess, expriming $x$ and $y$ like this : $x \in [f(y), g(y)], y \in [a, b] : b \gt a, g(y) \gt f(y) \ \forall y \in [a, b], \ g, f$ continuous, I have to ...
0
votes
2answers
40 views

Does $f(x) = -\sin(2x)$ have two integrals?

I found $\cos^2(x)$ and $\sin^2(x)$ which happily differ by the constant of $1$ though I've also found $\frac 12 \cos(2x)$, which both of the former diverge from by a sinusoidal function, what's wrong ...
6
votes
3answers
114 views

Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$

I can't solve the integral $$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$ I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.
-1
votes
0answers
49 views

How should I calculate $\int_a^b x \cdot \sin x \cdot \sqrt{1-x^2}\, dx$? [on hold]

Could anyone please show me how to calculate $$ \int_a^b x \cdot \sin x \cdot \sqrt{1-x^2}\ dx? $$
1
vote
0answers
30 views

How does the step in the picture transition to step 2?

:) I have a math question regarding this picture. The problem is that I do not understand how the first equation turns into the the second. Where did the integral come from?? (the dv and dt) Update: ...
0
votes
1answer
44 views

Does the integral converge

How to prove that the following integral doesn't converge? $$\int_0^\infty \frac{1}{(\ln^4x + \ln^2x)\ln^2(1-x^{1/3})^2(x + \sqrt{x} + 1)}dx$$ I suppose it doesn't converge because of quick growth ...
11
votes
1answer
89 views

Integral with arithmetic-geometric mean ${\large\int}_0^1\frac{x^z}{\operatorname{agm}(1,\,x)}dx$

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of positive numbers $a$ and $b$ is denoted $\operatorname{agm}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
1
vote
1answer
88 views

Please Verify the correctness of $\int_{0}^{\frac{\pi}{2}}\frac{x^2}{x^2+\ln^2(2\sin(x))}dx=\frac{\pi}{8}\left[\frac{\zeta(2)}{2}+\ln(2\pi)\right]$

I got the idea from here (1) $$\int_{0}^{\frac{\pi}{2}}\frac{x^2}{x^2+\ln^2(2\sin(x))}dx=\frac{\pi}{8}\left[\frac{\zeta(2)}{2}+\ln(2\pi)\right]$$ I am not quite sure it corrects, because I check it ...
1
vote
3answers
104 views

If a Riemann integrable function is zero on a dense set, then its integral is zero

Let $g:[a,b]\to\mathbb{R}$ be a Riemann-integrable function such that $g(x)=0$ for all $x\in A\subseteq[a,b]$ where $A$ is dense set. Then $$\int_{a}^{b} g=0$$ How can I show this?
1
vote
1answer
48 views

Find antiderivative of $ln(x)^y$ for any real y

What is this antiderivative? I have tested several values of $y$ in an online antiderivative calculator, but it's not clear how they are related. Here $y$ is fixed and I want the antiderivative with ...
0
votes
2answers
32 views

Extend a function 2pi periodically and calculate fourier

I have the function $$f(x)= \begin{cases} \frac{\pi}{2}+x & x \in (-\pi,0] \\ \frac{\pi}{2}-x & x \in (0,\pi]\\ \end{cases} $$ I need to extend it $2\pi$ periodically and then ...
-2
votes
0answers
54 views

How to integrate ${\int_0^{\pi} \frac{\sin (x)}{A-B\sin^3( x)} dx}$ [on hold]

$${2\over\int_0^{\pi} \frac{\sin x}{A-B\sin^3 x} dx} =\text{?}$$
-2
votes
1answer
48 views

Is there another approach to prove (1) without using substitution method? $\int_{0}^{\infty}\frac{1}{1+\phi^{nx+1}}dx=\frac{1}{n}$

$n\ge1$ $\phi=\frac{1+\sqrt5}{2}$ Is there another approach to prove (1) without using substitution method? (1) $$\int_{0}^{\infty}\frac{1}{1+\phi^{nx+1}}dx=\frac{1}{n}$$ Substitution method ...
1
vote
1answer
26 views

Multiple choice differential equation question.

Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuously differentiable function satisfying $$y(t)=y(0)+\int_{0}^{t}y(s)ds,t\geq 0$$ Then $1.y^{2}(t)=y^{2}(0)+\int_{0}^{t}y^{2}(s)ds$ ...
1
vote
0answers
45 views

what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
4
votes
3answers
253 views

Integration theorem: enough assumptions?

Let $f:[a,b]\to\mathbb{R}$ be a continious function. Show that if $$\int_a^b f(x)g(x)dx=0$$ for all continious functions $g:[a,b]\to\mathbb{R}$ with $g(a)=g(b)=0$, then $f(x)=0$ $\forall ...
7
votes
3answers
99 views

Is really $f(x)=\int g(x) dx$ a function?

I saw many of this kind of questions on some text/question books. Is there any other explanation of this, or is it really wrong as I thought? Here is a question of that kind: If $\displaystyle ...
1
vote
0answers
28 views

Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
0
votes
2answers
22 views

Fourier function expansion for extension over a $2\pi$ period

So I am currently looking at a fourier expansion for $$f(x)=\left\{\begin{array}{ccl}\sin x &\text{ if }& x\in[0,\pi]\\0 & \text{ if } & x\in[\pi,2\pi]\end{array}\right.$$ I am ...