All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
3
votes
0answers
182 views
About Henstock integrable vector-valued function
In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following ...
3
votes
2answers
60 views
Integral dx term sanity check
In my lecture slides, the dx term occasionally precedes the function being integrated. I understand that dx means "an infinitely small width of x" but does the definition of integration require it to ...
7
votes
5answers
97 views
Find the following integral: $\int {{{(\ln x)}^2}} dx$ by using the method of integration by parts
Find $\int {{{(\ln x)}^2}} dx$ by using the method of integration by parts.
My attempt:
$$\eqalign{
& \int {{{(\ln x)}^2}} dx = \int {2\ln x} dx \cr
& u = \ln x,{\rm{ }}{{du} \over ...
2
votes
1answer
31 views
Integrating a partial fraction with multiple quadratic denominators
When integrating a real rational fraction, you first break it into partial fractions. You then end up with fractions with linear denominators $\frac{A}{(x-b)^n}$, which are easy. You also end up with ...
2
votes
1answer
41 views
Breaking a contour integral into 3 separate contours?
We can try to integrate the following function around a counter-clockwise circular contour:
$$\frac{x^3}{(x-1)(x-2)(x-3)}$$
Can someone show how to use the Cauchy–Goursat theorem (explained here and ...
2
votes
1answer
46 views
Change of variables in a complex integral
I want to evaluate this integral using Residue Theorem
$$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$
$$ C : |z| = 1 $$
so I substitute letting $$\ W = z ^ {2 } $$
$$ dw = 2z dz $$
and the ...
3
votes
2answers
52 views
Evaluate $\int {\sin 2\theta \over 1 + \cos \theta} \, d\theta $, using the substitution $u = 1 + \cos \theta $
Evaluate $\int_0^{\pi \over 2} {{\sin 2\theta } \over {1 + \cos \theta }} \, d\theta $ using the substitution $u = 1 + \cos \theta $
$$\eqalign{
& \int_0^{\pi \over 2} {\sin 2\theta ...
4
votes
2answers
81 views
Find the following integral (most likely substitution)
$$\int_0^1 \frac{\ln(1+x^2)}{1+x^2} \ dx$$
I tried letting $x^2=\tan \theta$ but it didn't work. What should I do?
Please don't give full solution, just a hint and I will continue.
5
votes
2answers
69 views
Definite trig integral [duplicate]
How do I evaluate:
$$\int_{0}^{\pi} \sin (\sin x) \ dx$$
I have seen a similar question here but can't find it.
0
votes
1answer
29 views
Integrable functions and their inverses
Is the following statement true?
Let $f(x)$ be an integrable function on $[a,b]$. Suppose $|f(x)| \geq 1$ on $[a,b]$. Then, $\frac{1}{f(x)}$ is also integrable on $[a,b]$.
1
vote
1answer
58 views
Integration of a certain function
Any suggestion on how to evaluate the following?
$$\int_a^b \Big(\frac{b-x}{x}\Big)^{\frac{1}{n-1}} dx$$
where $0<a<b<+\infty$ and $n\geq 2$ is an integer.
Any idea for substitution or ...
2
votes
2answers
52 views
Evaluate $\int_0^{\pi \over 3} \sec x\tan x\sqrt {\sec x + 2} \, dx $ using a substitution of your choice
My attempt:
$$\eqalign{
& \int_0^{\pi \over 3} \sec x\tan x\sqrt {\sec x + 2} \, dx \cr
& u = \sec x \cr
& {du \over dx} = \sec x\tan x \cr
& {dx \over du} = {1 \over ...
4
votes
3answers
98 views
How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then
$$
\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0
$$
My attempt: I ...
0
votes
0answers
44 views
Simpson's rule characteristics
I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that:
...
1
vote
2answers
55 views
Find the integral of $\int {{2 \over {\sqrt x (x - 4)}}dx} $ given the substitution: $u = \sqrt x $
My stab at it:
$\eqalign{
& \int {{2 \over {\sqrt x (x - 4)}}dx} \cr
& u = \sqrt x \cr
& {{du} \over {dx}} = {1 \over 2}{x^{ - {1 \over 2}}} = {1 \over {2\sqrt x }} \cr
...
2
votes
1answer
63 views
How can I find this elliptic integral?
$$\int \frac{1}{\sqrt{1+x^{4}}}$$
After trying to do it for a while I posted it on Wolfram Alpha but I get no solution. How do I do it?
5
votes
1answer
51 views
What are some difficult integrals done by substitution and elementary functions?
What are some examples of difficult integrals that are done using substitutions?
For example: $$\int{\frac{(1+x^{2})dx}{(1-x^{2})\sqrt{1+x^{4}}}}$$
Please no laplace and fourier transforms as I ...
2
votes
2answers
62 views
evaluate $\int\ln x\tan x\,dx$
How to evaluate $\int\ln x\tan x\,dx$ ?
I've tried to do integration by parts but after calculations it cancel out the main question.
5
votes
1answer
75 views
Find the following integral: [duplicate]
Find $$\int \sqrt{\tan x}dx$$
My attempt:
$$\text{Let}\ I=\int \sqrt{\tan(x)}dx$$
$$\text{Let}\ u=\tan(x), du=(1+\tan^{2}(x))dx$$
$$I=\int \frac{\sqrt{u}}{u^{2}+1}$$
$$\text{Let}\ v=\sqrt{u}, ...
0
votes
2answers
34 views
Absolutely convergence of an integral
Let $f(x)=\dfrac{cos(x)}{x}$ and $I=\int_1^\infty f(x)dx$:
a. Does $I$ absolutely convergent?
b. if not, does it conditionally convergent?
I checked in matlab and it seems that a is incorrect but b ...
2
votes
1answer
40 views
Trigonometric Integration + Series
I am doing an integration question:
$$\int \frac{1-\cos^{2m}x}{1-\cos^2x}$$
So I have to show that $$\frac{1-\cos^{2m}x}{1-\cos^2x}=1+\cos^2x+\cos^4x+...+\cos^{2(m-1)}$$
How can I do that?
1
vote
3answers
34 views
Trapezoidal Rule/ Having trouble understanding what is k?!
QUESTION:
I know of the more understanding formula for the trapezoidal rule. However, I came across this new form in a book i'm reading. Can someone tell me how I'm suppose to enter the respected ...
2
votes
3answers
72 views
An easier way to find the integral of: $\int {x\sqrt {2 + x} {\rm{ }}dx} $, where ${u^2} = 2 + x$
My attempt at the question:
$\eqalign{
& \int {x\sqrt {2 + x} {\rm{ }}dx} \cr
& {u^2} = 2 + x \cr
& 2u{{du} \over {dx}} = 1 \cr
& {{du} \over {dx}} = {1 \over {2u}} ...
0
votes
1answer
79 views
Limit of $f_n(x)=\int _0^x\frac{1}{(e^t+e^{-t})^n}\,dt $
We suppose the function defined as $\forall n \in \mathbb{N}\setminus{0} \quad f_n(x)=\int _0^x\dfrac{1}{(e^t+e^{-t})^n}\,dt $, and suppose it has a limit $\lambda_n$ at $+\infty$.
The questions are:
...
1
vote
1answer
40 views
Integration Real Analysis
Let $E=\{1/n:n\in\mathbb{N}\}$ and consider the function on $[0,1]$ defined by $$f(x)=\begin{cases}\,1, &x\in E\\\,0,&\text{otherwise}\end{cases}.$$
Prove that $f$ is integrable on $[0,1]$ ...
1
vote
2answers
37 views
How to use the substitution rule for indefinite integrals to obtain the following result?
My book claims the following: $$ x = f(t) $$ $$y = g(t)$$
then, by substitution rule $$ \int y \ dx = \int g(t)f'(t) \ dt$$
I cannot find a way to obtain this result. Could someone show all the ...
7
votes
6answers
145 views
Find the following integral: $\int {{{1 + \sin x} \over {\cos x}}dx} $
My attempt:
$\int {{{1 + \sin x} \over {\cos x}}dx} $,
given : $u = \sin x$
I use the general rule:
$\eqalign{
& \int {f(x)dx = \int {f\left[ {g(u)} \right]{{dx} \over {du}}du} } \cr
...
0
votes
2answers
43 views
How to calculate the volume of the Hyperboloid using integrals?
I would like to calculate the volume of a hyperboloid described by this equation: $x^2 + y^2 - z^2 \leq 1$. I made some calculation: $x = \sqrt{1+z^2}$. But I don't know how to continue.
6
votes
3answers
76 views
Help for solving this integral please
The question :
$$ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $$
I tried dividing by $\cos^2 x$ and splitting the fraction.
That turned out to be complicated(Atleast for me!)
Help plz!!
4
votes
2answers
59 views
Calculating integral with branch cut.
I'm learning how to calculate integrals with branch points using branch cut. For example:
$$I=a\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{(1+\xi^{2})\sqrt{\frac{2}{m}\left(E-U_{0}\xi^{2}\right)}}$$
where ...
9
votes
3answers
104 views
Need help to evaluate this integral: $\int \frac {dx} {2x \sqrt{1-x}\sqrt{2-x + \sqrt{1-x}}}$
$$\int \frac {dx} {2x \sqrt{1-x}\sqrt{2-x + \sqrt{1-x}}}$$
Hey there, I've got this complicated integral to evaluate, but I don't know how to go about. I have tried making two substitutions:
$ t^2 ...
1
vote
2answers
29 views
Existence of limit of $\lim_{n \to \infty}\sum_{i=0}^{[n/2]} \frac 1 n f \left(\frac i n \right)$!
If $f$ is continuous in $[0,1]$ then $$\lim_{n \to \infty}\sum_{i=0}^{[n/2]} \frac 1 n f \left(\frac i n \right)$$ (where $[y]$ is the largest integer less than or equal to $y$)
(A) does not ...
2
votes
1answer
79 views
Can somebody provide an explanation to the formula of a one elementary integral?
Here is the formula:
$$
\int{\frac{dx}{x}} = \ln{|x|} + C
$$
In my textbook it is given without proof, so I have a little confusion here. From the definition of integral this equality must be true:
...
1
vote
1answer
50 views
Laplace equation and integral
$$ \int_0^{2\pi} \frac{1+3 \sin{\phi}}{a^2-2ar \cos(\theta - \phi) + r^2 } d\phi$$
Help me plz ... I have tried to solve this. but I still don't know.
0
votes
1answer
74 views
This is a formula in my text book,but I doubt the validity of it.
This is a formula in my text book,but I doubt the validity of it.If it is right,please give me a more detailed derivation.If it is wrong,please give me a right answer.
1
vote
2answers
84 views
It is an easy question about integral,but I need your help. [closed]
How to compute this integral?
$$ \int^{\pi}_{0} \frac{\sin(nx)\cos\left ( \frac{x}{2} \right )}{\sin \left ( \frac{x}{2} \right ) } \, dx$$
I need your help.
4
votes
1answer
66 views
Square integrable function that doesn't go to zero?
I'm reading through some elementary quantum mechanics textbooks and a few authors mention that there are functions that are "there exist pathological functions that are square-integrable but do not go ...
0
votes
2answers
51 views
Does $\frac 1{z^2+1}$ have a primitive on $\mathbb C-\{i,-i\}$?
Please explain why the above is False. (I do not understand what the hint is trying to say either: the Cauchy Integral Theorem that I know states that If $f$ is analytic on a simply-connected domain ...
4
votes
1answer
45 views
Integrate using partial fractions, answer discrepancy
My attempt:
$\eqalign{
& \int {{{{x^2} + x + 2} \over {3 - 2x - {x^2}}}} dx \cr
& {{ - 1( - {x^2} - 2x + 3) - 2x + 3 + x + 2} \over {3 - 2x - {x^2}}} \cr
& = - 1 + {{ - x + 5} ...
0
votes
1answer
54 views
Can we combine nested integrals into one integral?
If we are given a double integral,
$$\int_c^d{\int_a^b{f(x,y) dx} dy}$$
can we convert this into a single integral, i.e.
$$\int_{a_2}^{b_2}{f(z)dz}$$
...where $a_2$ and $b_2$ can be whatever makes ...
5
votes
3answers
111 views
What does $ d\tan(x) = \sec^2(x)dx $ mean?
What does $ d\tan(x) = \sec^2(x)dx$ mean?
I've seen it used in integration problems to make it more simpler. However, I'm not really sure what it means. Can someone explain this to me?
1
vote
1answer
30 views
integration question very short help me?>
I am not sure if i should solve this with integration by partial fracions
So,Im in the middle of solving an exercise but I got here $-u^3/(2+2u^2) $..
how to integrate this by partial fraction?
how ...
2
votes
2answers
57 views
Beta Function : Proof
Prove that
$$\int_{0}^1 \frac{x^{p-1} + x^{q-1}}{(1 + x)^{p+q}} dx = \beta (p,q)$$
Please help !
1
vote
3answers
81 views
Integrating a school homework question.
Show that $$\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \frac{a\sqrt{3}+b-\pi}{6},$$ where $a$ and $b$ are constants to be found.
Answer is: $$\frac{24\sqrt3-48-\pi}{6}$$
Thank you in advance!
13
votes
3answers
218 views
Proving a trig infinite sum using integration
How can I prove the following using integration and elementary functions?
Prove that:
$$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$
$0 < \theta < 2\pi$
0
votes
1answer
56 views
Gamma and Beta Functions : Integration
Evaluate the integral :
$$\int_0^ax^3(a^3-x^3)^5dx$$
Thanks in advance.
-1
votes
0answers
34 views
Please help me to solve this Henstock Integration [closed]
using cauchy criterion, show that the following functions are Henstock Integrable or not
a.$$f:[0,1]\to R \text{ where }f(x)=
\begin{cases}
1,\quad &x \text{ is rational}\,\\
0,&x \text{ is ...
-2
votes
2answers
45 views
prove $ e^{bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds $ [duplicate]
please How can I prove that
$$
e^{-bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds
$$
f non-negative measurable function
I would appreciate it enormously if ...
0
votes
1answer
47 views
prove that integral
prove that
$$-\int_0^t \ sgn(f({s})) \ d{s}=\int_0^t \ sgn(-f({s}))\ d{s}+2\int_0^t 1_{f({s}) =0}d{s}$$
with
$$ sgn(x) := \begin{cases}
-1 & \text{if } x =< 0, \\
1 & \text{if } x > ...
0
votes
1answer
35 views
integral convergence - does this integration converge
i just ran into this problem and i'm having a hard time solving it:
i would like to know if this integral converges or not, and why.
i'd prefer the normal convergence tests.
$$
...
