All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
2
votes
1answer
40 views
How should I numerically solve this PDE?
I am hoping to figure out the function $u(x,y,t)$ for some integer arguments when $u(x,y,0)$ is given (by figuring out I mean generating some images in MatLab), also time $t \ge 0$.
$$\frac{\partial ...
2
votes
2answers
72 views
Prove $\frac{1}{2 \pi} \int_0^{2\pi} \frac{1 - r^2}{1 + r^2 - 2r \cos{t}} dt = 1$ using contour integration
The question is to solve the integral using concepts of contour integrals:
$$\frac{1}{2 \pi} \int_0^{2\pi} \frac{1 - r^2}{1 + r^2 - 2r \cos{t}} dt = 1$$
4
votes
1answer
59 views
calculate $ F(x)= \int_{0}^{\sin x}\sqrt{1-t^2}\,dt $
Calculate $F'(x)$
I have this exercise in my worksheet, I am having a problem obtaining the correct answer which is as listed on the answer sheet $-\cos^2(x)$.
4
votes
3answers
111 views
How to find $\int_0^{2\pi} \frac{dt}{1+2\cos(t)}$
The problem is
$$\int_0^{2\pi} \frac{dt}{1+2\cos(t)}.$$ I know it is equal to
$$\int\limits_{\text{unit circle}}\frac{2dz}{i(1+z)^2}$$
but I don't know how I should calculate the last integral.
1
vote
4answers
87 views
How to integrate $\int e^{-x}\sin(3x)\;dx$?
I want to integrate following by the method of integration by parts
$$\frac{\cos(3x)}{e^{x}}$$
when I try to solve it by integration by parts it always leads to something like as mentioned below ...
0
votes
1answer
26 views
The inequality of an integration
Let $f(x) \le 0$ with $x \in [0,\,x_0]$ and $f(x)>0$ with $x>x_0$. We also assume $\alpha(x)<g(x)<\beta(x)\le0$ with all $x\ge0$. Is the following inequality true ?
$$
\int_0^\infty {f(x) ...
0
votes
2answers
27 views
Help on this integral
I'd like to know why this holds
If one have $f(x_t,t)=x_t\mathrm{e}^{\theta t}$
$\int_0^tdf(x_t,t)=x_t\mathrm{e}^{\theta t}-x_0$
That shouldn't be only $x_t\mathrm{e}^{\theta t}$?
6
votes
0answers
94 views
Help on an integral.
I have to finish following integral
$$\int_0^{+\infty} \log_2(1+a x) (1-e^{-\frac{x}{2}})^{b-1} \frac{1}{2}e^{-\frac{x}{2}} dx$$
After a week of work on this, I found it may be impossible to have a ...
1
vote
1answer
38 views
How To Set The Area Between Two Functions Equal To A Constant?
Please pardon the broad nature of this question. Suppose two functions encompass an area between them. What approach might be taken to adjust either function through adding constants to set the area ...
1
vote
0answers
27 views
Definite integral involving modified bessel function of first kind, exponentials, and powers
Is there any solution for this integral?
$$
\int_a^b xe^{-\alpha x^2}J_n(\beta x) dx\,.
$$
I looked up in all books i could find and only found this:
$$
\int_0^{\infty} xe^{-\alpha x^2}J_n(\beta x) ...
0
votes
1answer
32 views
Difference of definite integrals inequality
Could you help me how prove that for any $\mathcal{C}^1$ function we have:
$$\left|\int_{a} ^{\frac{a+b}{2}}f(x) d x - \int_{\frac{a+b}{2}} ^bf(x)dx\right| \le \frac{(b-a)^2}{4} \cdot \max _{x \in ...
0
votes
1answer
33 views
How do I calculate the complex integral $\int_c\frac{1}{z^2}\,\mathrm{d}z$?
How do I calculate the complex integral $\displaystyle\int_c\frac{1}{z^2}\,\mathrm{d}z$, where $c$ is a direct line from $z=0$ to $z=1+2i$?
Is it possible solving such the integral? I mean ...
0
votes
2answers
24 views
Solving the next Integral (by parts?)
Can anyone give me a hint / help with the next integral? Thanks!
$$\displaystyle\int_{0}^{t}{x^{a-1}(t-x)^{b-1}dx}$$
1
vote
1answer
49 views
Lebesgue vs. Riemann integrable function
While trying to learn the difference between Lebesgue and Riemann integrals, I came across the following example:
$$\int_{0}^{1}t^\lambda\,\mathrm dt$$
What I know so far:
only for $\lambda>0$ ...
3
votes
2answers
25 views
integrate product of trig functions
I need to find the Fourier cosine series for $\cos(3x)\sin^2(x)$, But I don't even know where to start to determine $$\int _0^{\pi }\cos(3x)\sin^2(x)\cos(k x)dx$$
2
votes
1answer
61 views
prove this $\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$
let $f\in C^1[0,2]$,and such $\int_{0}^{2}f(x)dx=0,f(0)=f(2)$,
show that
$$\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$$
I think we must use $Cauchy$ inequality
my idea:I have see this
let ...
1
vote
1answer
34 views
If $\int^{1}_0 \frac{tan^{-1}x} {x} dx = k \int^{\pi/2}_0 \frac {x}{sinx} dx$, find $k$.
Problem :If $\int^{1}_0 \frac{tan^{-1}x} {x} dx = k \int^{\pi/2}_0 \frac {x}{sinx} dx$, find $k$.
Solution : If we put $x=tant$ in $\int^{1}_0 \frac{tan^{-1}x} {x} dx$
then integral ...
2
votes
1answer
54 views
The notion of a curve in the context of line integrals
For brevity I'm making the following assumption: I'm only talking
about regular curves on $\left[a,b\right]$ with values in $\mathbb{R}^{n}$,
and line integrals of scalar fields.
[Since there are a ...
4
votes
0answers
127 views
A difficult integral $\int_0^\infty \mathrm{d}t\frac{1}{t}\frac{1}{t-s-\mathrm{i}\epsilon}\frac{1}{X}\ln\frac{1-X}{1+X} $
Can anyone give any hints on how to rewrite this in terms of dilogarithms?
$$\int_0^\infty ...
3
votes
1answer
51 views
What is meant by $|dxdy|^{1/2}$ in the integral?
In this Daniel Grieser - Basics of the b-calculus paper the author mentions the term of a half-density on page 54 as an object which look like $u(x) |dx_1 \cdots dx_n|^{\frac{1}{2}}$. And I'm not ...
5
votes
1answer
136 views
+50
Does this integral have a closed form: $\int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y}$?
Consider the following integral:
$$G(\beta,\delta,y) = \int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y},$$
with $\delta>0$, $\Re\beta>0$, $y\neq1$.
Does it have a closed form in ...
1
vote
1answer
57 views
Differentiable functions without an antiderivative
Specifically, why is there no antiderivative, or any possible method of integrating (except numerically) say $\;e^{\csc(x)}$?
(I don't have my computer handy right now so I cant format the formula, ...
1
vote
0answers
21 views
Generalized Riemann Integral
Is there any usage of studying the Henstock-Kurzweil
integral as such ?
It doesn't seem to be as popular a method of integration as the Lebesgue integral or even the Riemann-Stieltjes ...
0
votes
1answer
59 views
How to solve integrals of type $ \int\frac{1}{(a+b\sin x)^4}dx$ and $\int\frac{1}{(a+b\cos x)^4}dx$
$$\displaystyle \int\frac{1}{(a+b\sin x)^4}dx,~~~~\text{and}~~~~\displaystyle \int\frac{1}{(a+b\cos x)^4}dx,$$
although i have tried using Trg. substution. but nothing get
2
votes
1answer
46 views
How can I devise a general approach to solving an indefinite integration problem?
INTRODUCTION:
I'm trying to create a general approach in solving integration problems around 7 specific methods.
Basic Formulas, Substitution, Numerical Integration, Integration by Parts, ...
3
votes
2answers
41 views
Volume integral of the curl of a vector field
I am having hard time recalling some of the theorems of vector calculus. I want to calculate the volume integral of the curl of a vector field, which would give a vector as the answer. Is there any ...
0
votes
0answers
28 views
About Stieltjes Sums
Let $[a,b]\subset \mathbb{R}$. A tagged partition of $[a,b]$ is a set
$D=\{(t_i,[x_{i-1},x_i])\}_{i=1}^m$ where $\{[x_{i-1},x_i]\}_{i=1}^m$ is a partition of $[a,b]$ and $t_i\in [x_{i-1},x_i]$; $t_i$ ...
3
votes
3answers
39 views
Uniform convergence of composition of functions and integration
I've been wondering about this problem for a bit, it came up in class but its not really homework.
Let $f:[0,1]->R$ be continuous and non-negative.
We know $f(x^n)\to f(0)$ for $x \in [0,1)$ and
...
4
votes
2answers
69 views
Evaluate : $\int^{\frac{\pi}{2}}_0 \frac{\cos^2x\,dx}{\cos^2x+4\sin^2x}$
Evaluate: $$\int^{\frac{\pi}{2}}_0 \frac{\cos^2x\,dx}{\cos^2x+4\sin^2x}$$
First approach :
$$\int^{\frac{\pi}{2}}_0 \frac{\cos^2x\,dx}{\cos^2x+4(1-\cos^2x)}$$
$$=\int^{\frac{\pi}{2}}_0 ...
16
votes
4answers
495 views
$\int^{1}_{0} f^{-1} = 1 - \int^1_0 f$
One more from hard to believe facts, which I'm curious why are true.
Let $f : [0,1] \rightarrow [0,1] $ is a continuous, monotonically increasing and surjective function
Then
$$\int^{1}_{0} f^{-1} ...
3
votes
2answers
89 views
How to prove that $\int^1_0 \frac{1}{x^x} dx = \sum^{\infty}_{n=1} \frac{1}{n^n} $? [duplicate]
The task is: Prove, that
$$\int^1_0 \frac{1}{x^x} dx = \sum^{\infty}_{n=1} \frac{1}{n^n}$$
I completly don't have an idea, how to prove it. It seems very interesting, I will be glad if someone ...
3
votes
1answer
36 views
Integration by substitution problem $x = C \sin(t)$.
For solving the integral:
$$
\int_a^b \sqrt{\alpha^2 - \beta^2 x^2} \, dx
$$
I've been taught to use $x = \frac{\alpha}{\beta} \sin(t)$ in order to get
$$
\frac{\alpha^2}{\beta} \int_{\arcsin(a ...
2
votes
2answers
47 views
Finding indefinite integral by partial fractions
$$\displaystyle \int{dx\over{x(x^4-1)}}$$
Can this integral be calculated using the Partial Fractions method.
4
votes
1answer
77 views
How to find the following indefinite integral?
$$ \int {dx \over {\sin^3 x+\cos^3 x}}$$
Can this integral be found by substitution or any other method such as complex number?
6
votes
0answers
72 views
Help in calculating the following integral $\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $
I was asked to calculate this:
$$\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $$
My idea was to change the integration limits to $|z|=1$ in the complex plane and to ...
1
vote
0answers
49 views
Riemann sums and integral
If the Riemann sums of a function $f$, defined on an arbitrarily tagged partition $P$, are positive, then will the integral also be positive for any arbitrary tagged partition whose norm is bounded?
5
votes
1answer
113 views
+50
proof of stokes theorem
I don't understand the "idea" of the following proof, aswell as some of the steps.
As i'm not sure about its "ways", im not editing it much and as such it might be in the wrong order. My sincere ...
0
votes
2answers
44 views
Not able to solve $\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $
If $p=\frac{7}{8}$ then what should be the value of $\displaystyle\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $
when $$g(x) = x \log x \quad \text{or} \quad g(x) = \frac{x}{\log x}? $$
...
1
vote
1answer
36 views
Contructing a $\delta$-fine tagged partition from the old ones
Let $[a,b]\subset \mathbb{R}$. A tagged partition of $[a,b]$ is a set
$D=\{(t_i,I_i)\}_{i=1}^m$ where $\{I_i\}_{i=1}^m$ is a partition of $[a,b]$ consisting of closed non-overlapping subintervals of ...
2
votes
2answers
58 views
Evaluate $ \int^4_1 e^ \sqrt {x}dx $
Evaluate $ \int^4_1 e^ \sqrt {x}dx $
solution:-
Here $1<x<4$
$1<\sqrt x<2$
$e<e^ \sqrt {x}<e^ 2$
$\int^4_1 $e dx$<\int^4_1 e^ \sqrt {x}dx<\int^4_1 ...
5
votes
4answers
158 views
Finding another way of doing this integral $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$
Problem :
Integrate : $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$
I have the solution : We can substitute $\sqrt{x}= \cos^2t$ and proceeding further,
I got the the answer which is ...
1
vote
2answers
43 views
Integration of $\int(2-x/2)^2dx$
Got an exam tomorrow and my head is no longer working. Could someone walk through the integration of this function
$$\int\left(2-\frac x2\right)^2dx$$
I understand integration by parts and stuff ...
0
votes
1answer
52 views
Calculus - Indefinite integration Find $\int \sqrt{\cot x} +\sqrt{\tan x}dx$ [duplicate]
Problem : Find $\int \sqrt{\cot x} +\sqrt{ \tan x}dx$
My Working :
Let $I_1 = \sqrt{\cot x}dx$ and $I_2 = \sqrt{\tan x}dx$
By using integration by parts:
Therefore , $I_1 = \sqrt{\cot x}.\int1 ...
5
votes
3answers
91 views
Evaluate $ \int^{ \pi/2}_{- \pi/2} \frac {1}{ 1+e^{\sin x} }dx $
Evaluate $ \int^{\pi/2}_{-\pi/2} \frac {1}{ 1+e^{\sin x} }dx $
Solution:
I think odd, even functions are of no use here.
Also we get nothing by taking $e^{\sin x} $ common in denominator.
...
1
vote
0answers
34 views
Let $X$ be an integrable RV with median $m$. Show that $m=\text{argmin}_a\mathbb{E}|X-a|$
I have tried to look for mistakes in these calculations until I turned blue, but still clueless. If anyone could help me pointing out the errors, I would be really grateful.
My apology for the long ...
1
vote
2answers
54 views
If $\int^{\pi}_0 x f (\sin x) dx = k \int^{\pi/2}_0 f (\sin x) dx$, find $k$.
Problem : If $\int^{\pi}_0 x f (\sin x) dx = k \int^{\pi/2}_0 f (\sin x) dx$, find $k$.
Solution : Period of sine function is $2 \pi$
I don't know whether we can use the period of this ...
3
votes
1answer
48 views
integrating logarithm or x raised to a power?
$\int\frac{15}{x}dx$ would be 15$\int\frac{1}{x}dx$ = $15\ln|x|+c$.
This seems like a silly question but I'm feeling exceptionally dense today. Why would you apply the logarithm rule, why wouldn't ...
1
vote
2answers
46 views
How to solve this stochastic integrals?
how can I solve these two stochastic integrals?
$$\int_0^T B_t\,dB_t$$
$$\int_0^T f(B_t)\,dB_t$$
where B_t is the BM.
Thank you very very much!
0
votes
1answer
44 views
Convergence of Fourier integral
I have noticed that the Fourier integral $\int_{-\infty}^\infty \mathscr{F}f(\omega) e^{i\omega t} d\omega $ of $f(t) = \mu(t) e^{-t}$, where $\mathscr{F}f(\omega) = \frac{1}{2\pi} ...
4
votes
2answers
74 views
Why can't you find all antiderivatives by integrating a power series?
if
$f(x) = \sum\limits^{\infty}_{n=0}\frac{f^{(n)}(0)}{n!}x^n$
why can't you do the following to find a general solution
$F(x) \equiv \int f(x)dx$
$F(x) = \int ...
