Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

0
votes
0answers
4 views

Is it possible construct a well-defined series through integration by parts?

I have a function $$G=\int_0^S\mathrm{d}x\,f(x)g(x)\mathrm{e}^{-\int_0^x\mathrm{d}z\,g(z)}~.$$ If $f(x)$ was unity, the above integral could have been easily written as $$G=1-\mathrm{e}^{-\int_0^S\...
0
votes
0answers
15 views

Integration of the following trignometry

We have to find the integration of the following , I tried but got stuck , can anyone help me
0
votes
2answers
45 views

Evaluate $\int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }}$

How do you solve this integral $$ \phi = \int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }} $$ ? Note: It appears in the Kepler problem and it should ...
1
vote
1answer
30 views

Why do I get two answer when calculating this integral from two ways?

Assuming $a(t)=a_0\sin(\omega t)$, $v(0)=0$ and $x(0)=0$. I hope you know about basic relation between position, velocity and acceleration. They are derivatives of the proceeding one. I went on ...
2
votes
0answers
27 views

integration by parts of multiple functions

Can anyone help me solve this problem? $I_0$ represents the modified Bessel function.Thanks. $$\int_{a-b}^{a+b}\frac{\exp\left(-\frac{c\,t^2}{2ab}\right)\,t\,I_0(t)}{ab\sqrt{1-\frac{(t^2-a^2-b^2)^2}{...
2
votes
3answers
72 views

Solving an equation involving an integral: $\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$

Determine a pair of number $a$ and $b$ for which $$\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$$ I tried putting $x$ as $1-x$ as the integral wouldn't change but could not move forward from ...
-1
votes
1answer
46 views

As I can find the critical points of the function $\int_a^b (y^{2}+2(y')^{2}+(y'')^{2}) dx.$ [on hold]

Hi I'm stuck with a problem, greatly appreciate a suggestion to solve: As I can find the critical points of the function $$\int_a^b (y^{2}+2(y')^{2}+(y'')^{2}) dx.$$
-4
votes
0answers
51 views

Is this integral equals zero? [duplicate]

I want to calculate this integral $$\int_{-\infty}^{\ln(4)}\frac{xe^x}{\sqrt{4e^x-e^{2x}}}dx$$ How do I calculate this integral?
4
votes
1answer
141 views

What is relation between these integrals

I know $$ \int_{0}^{\frac{\pi}{2}}\ln(\sin x)dx=-\frac{\pi}{2}\ln(2)$$ What is relation between it and $$\int_{-\infty}^{\ln(4)}\frac{xe^x}{\sqrt{4e^x-e^{2x}}}dx$$ Please guid me. I have sixteen ...
0
votes
0answers
41 views

On the vanishing of integrals involving the $\sinh$ function. [on hold]

Suppose for some positive real $\theta$ that $$\int_1^\infty f(x)\sinh(\theta\log \sqrt x) \mathrm{d}x = 0$$ Where $f(x)$ is a non-constant and continuous function of $x$. What necessary properties ...
0
votes
0answers
12 views

Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want to ...
13
votes
1answer
232 views
+50

A tricky integral - $\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt $

$$ \mathbf{\mbox{Evaluate:}}\qquad \int_{0}^{1} \sqrt{\frac{1}{\left(1 - t^{2}\right)^2} - \frac{\left(n + 1\right)^{2}\,t^{2n}}{\left(\, 1 - t^{2n+2}\,\,\right)^{2}}} \,\,\mathrm{d}t $$ where $n$ ...
3
votes
3answers
127 views

Help on how to show that $\int_{0}^{1}\left(2{x-1\over \ln^2{x}}-{x+1\over \ln{x}}\right)dx=3\ln{2}-2$

$$\int_{0}^{1}\left(2{x-1\over \ln^2{x}}-{x+1\over \ln{x}}\right)dx=3\ln{2}-2\tag1$$ Rewrite, so we can apply Frullani's formula on first part $$\int_{0}^{1}\left(-{x+1\over \ln{x}}+{2\over \ln{x}}+{...
0
votes
1answer
20 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
1
vote
0answers
33 views

Is there a general method to go about deriving a definite integral for a given result?

I was reading a blog post earlier about the Sophomore's Dream and a question came to mind: Say we wanted to find a definite integral that gives the following result $$\sum_{n=1}^\infty \left(\frac{a}...
18
votes
3answers
287 views

$\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that?

If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have: $$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} f(x) dx \tag{1}$$ The ...
-2
votes
0answers
36 views

ratio of 2 definite integrals [on hold]

IfA=∫(sin(884x)sin(1122x)/sin(x), x, 0, pi) and B=∫(x^238(x^1768-1)/(x^2-1), x, 0, 1) then the value of A/B? Since in A we can write it as ∫(sin(884x)sin(1122x)/sin(x), x, 0, pi/2)and then x-->π/2-x ...
0
votes
1answer
31 views

Integral Convergence with parameters

I am finding it hard to approach this question: $$\int_0^{\pi/2} {1-\cos(x)^a\over x^b}\, dx$$ and I need to determine for which positive values of $a,b$ the integral converges. Thanks,
2
votes
2answers
40 views

Area of the polar figure enclosed by the circle $r=2$ and the cardioid $r=2(1+cos θ)$

This is exercise 7, of the book Engineering Mathematics by Stroud, Chapter 24, Further Problems section. Here's a graph i made of the figure as i see it: It gives the answer as $π+8$. The integral ...
0
votes
0answers
28 views

Proving that an integral of several cdf and pdf functions is increasing in a certain parameter.

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
-1
votes
0answers
17 views

Calculate integral $x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}$ [on hold]

Calculate $\int_0^{1-x2}\ x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}dx_1$ If $n,l,k$ is fixed -it's easy - just expand polynomial and cause similar terms. But what I can do in this common case?
1
vote
0answers
27 views

How to write/calculate an integral that covers $x\in\mathbb C$?

Consider some function $f(x)$ defined for $x\in\mathbb C$, and it is integrate-able as well. If I wanted to calculate the 'area' under the graph for $x\in\mathbb R$, I could use $\int_{-\infty}^\...
2
votes
2answers
57 views

Summing a series of integrals

I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here. I have a series of integrals I would like to sum, but I don't ...
4
votes
3answers
398 views

Integration by parts: is this legitimate way of using?

Is it legitimate to write $$\int_0^a\mathrm{d}x\,f(x)g(x)=\left[f(x)\int_0^x\mathrm{d}x\,g(x)\right]_0^a-\int_0^a\mathrm{d}x\,\frac{\mathrm{d}f(x)}{\mathrm{d}x}\int_0^x\mathrm{d}x\,g(x)$$ Thanks.
0
votes
2answers
66 views

Maxwellian integral : is there a closed form?

$f_A(x,y)=\int_0^\infty du \frac{u \left(e^{-\frac{(u-x)^2}{2 A}}-e^{-\frac{(u+x)^2}{2 A}} \right)}{\sqrt{2 \pi } \sqrt{A} x \left(y^2+u^2\right)} $ is there a closed form? I was able to find ...
-1
votes
0answers
56 views

Evaluate $\int_0^\frac12 \frac{\sin(\pi x)}{(x+1)(x+2)} dx$ [closed]

$f(x) = \int_0^\frac12 \frac{\sin(\pi x)}{(x+1)(x+2)} dx$ Could not solve the problem. Can anyone help me ?
-1
votes
0answers
14 views

Sum of Independent Levy RVs is Levy RV [closed]

I want to show that the summation of independent Levy random variables X and Y with scaling parameters a and b is a Levy random variable with scaling parameter c = (a^(1/2)+b^(1/2))^2 using ...
2
votes
1answer
76 views

Integrating $e^{a\cos(\phi_1-\phi_2)+b\cos(\phi_1-\phi_3)+c\cos(\phi_2-\phi_3)}$ over $[0,2\pi]^3$

I am trying to integrate the following function. (it arises in channel modeling in wireless communications, Rayleigh random variables)..Any help is appreciated.Thanks $$\int_0^{2\pi}\int_0^{2\pi}\...
1
vote
3answers
97 views

Show that if a function is nonnegative and continuous on [a,b] then f(x)=0 for all x in [a,b] [duplicate]

Show that if $f : [a, b] \to \mathbb{R}$ is non-negative and continuous on $[a, b]$ and $$\int_{a}^{b} f(x)dx = 0$$ then $f(x)=0$, for all $x ∈ [a, b]$. I'm having difficulties in proving things to ...
1
vote
2answers
73 views

Calculate the value of $e$ from integral definition

Starting with the definition of $e$ as $$\int_1^e \frac{dx}{x} = 1,$$ how can I show that $e = 2.718\ldots$?
0
votes
2answers
32 views

Changing Integral Bounds

I'm studying for Exam P and I was wondering what the need to change the lower limit of this integral was. Substituting $u=1+x$, $du=dx$, $$ \int_0^\infty\frac{3x}{(1+x)^4}\;dx = \int_1^\...
6
votes
1answer
55 views

Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$

There are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, ...
6
votes
4answers
197 views

Why is $\int_0^3 \frac{1}{\sqrt{x-3}}\, \mathrm{d}x$ a complex number?

This is my first question on this site, so please pardon any nuanced formatting errors. My friend and I were discussing the following integral yesterday: $$\int_0^3 \frac{1}{\sqrt{x-3}} \, \mathrm{...
6
votes
4answers
146 views

A series with logarithms

Can we express in terms of known constants the sum: $$\mathcal{S}=\sum_{n=1}^{\infty} \frac{\log (n+1)-\log n}{n}$$ First of all it converges , but not matter what I try or whatever technic I am ...
-2
votes
1answer
42 views

A question on the Laplace Transform of $f(t)=t e^{at}\sin (bt)$ [closed]

I would like to solve the Laplace transform of the following function: $$t \mapsto t e^{at}\sin (bt).$$ I know that $\mathscr{L}\left(\sin(bt)\right)=\dfrac{b}{s^2+b^2}$ and that you have to ...
0
votes
1answer
23 views

Area included in the graph of various functions

I've some problems recognizing which one is the area between various function. In this case i need to calculate the area between 3 lines and a curve, exactly between: $x+3,x^2-9,x=5 ,x=0$ I can't ...
0
votes
3answers
61 views

Setting Up a Double Integral Over a Region

Evaluate $$\iint_R(x+y)\ \mathrm dy\ \mathrm dx$$ where the region $R$ is bounded by $y=\frac{1}{9}x, x=6$ and the x−axis. can anyone show me how to set this up? I thought that it would integrated ...
3
votes
2answers
116 views

Understanding Integration

This is a question coming from a Math newb. When I learn something I try to connect it to what I already know, so please bear with me. I know how to calculate integrals, but I never really understood ...
0
votes
1answer
53 views

Can I solve this integral problem?

I want to know how can i solve this function. $\int{(1-y^d)^n}dy$ Is it possible to solve it? If you know the method, please teach me.
8
votes
0answers
140 views

Closed form of $\int_0^1 \tan(\gamma\sqrt{1-x^2}) dx$

Some context: I'm studying the problem of nonperturbative pair creation from strong fields in quantum electrodynamics. For certain time dependent electric fields I can get some information about the ...
1
vote
3answers
42 views

Area between four functions

I'm not sure about calculating the area between: $f(x)=x+3$ ,$g(x)=x^2-9$,$k(x)=5$ and $y=0$ My idea , but i'm not sure about the first integral, is: $$$$ $\int_\sqrt{14}^05-(x-3)dx$ + $\int_0^25-(...
0
votes
0answers
31 views

Simplify an Integration with cosine, logarithm and hyperbolic functions

I would like to simplify (or solve) the following integral: $$F(t):=\int_{a}^{t-a} \cos\left(\ln\left( \frac{f(t-x)}{f(x)} \right)\lambda \right)\frac{1}{\sqrt{f(t-x)\cdot f(x)}} dx, \quad t>2a,$$ ...
21
votes
3answers
464 views

The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$

I hope you find this integral interesting. Evaluate $$\int_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right) \sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\...
22
votes
6answers
1k views

Conjectured value of $\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$

I was curious whether this integral has a closed form expression : $$\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$$ The integrand has a singularity ...
1
vote
1answer
41 views

Find $f$ such that $\int^b_a f^2(x) = c$ and $\int^b_a f(x)g(x) dx$ should be maximum

Given $g$ integrable on $[a,b]$, find integrable $f$ such that $\int^b_a f^2(x) = c > 0$ and $\int^b_a f(x)g(x) dx$ should be maximum. I tried to use Cauchy–Schwarz inequality and define $d:= \...
1
vote
3answers
97 views

Evaluation of $4\int_0^{+\infty} \frac{\left(\sinh\left(\frac{x}{8}\right)\right)^2}{x(e^x-1)}dx$

In relation with Evaluating series of zeta values like $\sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=\ln(\pi)-\frac{3}{2}\ln(2) $ From the well-known formula, For $s$, such that $\Re(s)>1$, $\...
-1
votes
0answers
21 views

Change the order of integral

How to change the integral $\int_a^y G(x)\int_a^x F(t) \int_a^t W(z) dzdtdx$ into the form $\int_{?}^{?} F(x) \int_a^{?} G(t) \int_a^{?} W(z)dzdtdx$
0
votes
0answers
11 views

Integral after applying monotonic function

Let $F$, $G$ are two cdf contained in [a,b], $w$ is a non-negative and strictly increasing probability weighting function. If $\int_a^b G(x)-F(x) dx =0$, and $\int_a^y P(x)\int_a^x G(z)-F(z)dzdx \geq ...
1
vote
1answer
31 views

Definite integral - Different answers while using different approaches to solve

I wanted to ask whether it is possible to get 2 different answers for the same definite integral, using two different approaches to solve it. My friend and I have received an exercise in which we ...
3
votes
1answer
47 views

Show the triple integral given is equivalent to $\frac{15\pi}{16}$

Evaluate $$\iiint_E\;z \, dV$$ where E is enclosed between the spheres $x^2 + y^2 + z^2 = 1$and$x^2 + y^2 + z^2 = 4$ in the first octant. I'll be honest. My first ...