All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
0
votes
1answer
23 views
Question on integration
Let $(p,q)\in \mathbb{R}^2$ , and $H: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$
if $q'=\frac{\partial H}{\partial p} (p,q) ~~\text{and}~~ p'=-\frac{\partial H}{\partial q} (p,q)$
How to ...
1
vote
1answer
23 views
Integration of volumes of revolution: bisector surface
$y = (16-x^2)^{0.5}$ is rotated around the $x$-axis to give a sphere of radius $4$ units. Find the equation of the straight line that passes through $(-4,0)$, such that when also rotated around the ...
1
vote
0answers
67 views
The integral of $\frac{1}{1+x^n}$
Motivated by this question:
Integration of $\displaystyle \int\frac{1}{1+x^8}dx$
I got curious about finding a general expression for the integral $\int \frac{1}{1+x^n},\,n \geq 1$. By factoring ...
1
vote
1answer
63 views
Show that $\int f(x)dx=\int g(u)du$
Dose anyone know how to do this?
If a function $f(x)$ can be written $f(x)=g(u(x))du(x)/dx$ for a suitable function $u(x)$, show that $\int f(x)dx=\int g(u)du$.
2
votes
2answers
35 views
Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$
I'm trying to evaluate the integral $$\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$$
where $\chi_{[0,1]}(x)=1$ is the characteristic function, i.e. equals $1$ for $x \in ...
3
votes
1answer
34 views
Using a definite integral, to create a specific recurrence relation.
Hello i have the integral:
$$y_n=\int_0^1\frac{x^n}{x+5}dx$$ where $ n=1,2,3,4,....,\infty$
I need to show that the integral can be represented by the recurrence relation below;
$$y_n= ...
2
votes
3answers
62 views
Vector calculus for ellipse in polar coordinates
I'm having trouble with this question, can somebody please help me with it! I'll thanks/like your comment if help me =)
I know that for a ellipse the parametric is $x=a\sin t$ , $b= b \cos t$, ...
12
votes
4answers
1k views
Why doesn't integrating the area of the square give the volume of the cube?
I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):
$$V = \int\limits_a^b {A(x)dx}$$
But this doesn't seem to work with the square. ...
1
vote
0answers
44 views
Antiderivative of an absolute function
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
2
votes
1answer
34 views
Leibniz rule, multiple integrals
Suppose I need to compute the derivative
$$
\frac{d}{dr} \int_{-\infty}^{\infty} \int_{h(r)}^\infty \int_{g(r)}^\infty {rf(x,y,z)\, dz\, dy\, dx}.
$$
Can I apply a Leibniz rule of some form? How?
2
votes
3answers
75 views
$\int^{\pi/2}_{0}\log|\sin x| \,dx = \int^{\pi/2}_{0}\log|\cos x| \,dx $
Prove that :
$$\int^{\pi/2}_0 \log|\sin x| \,dx = \int^{\pi/2}_0 \log|\cos x| \,dx $$
I tried to cut the integral into a sum of parts and changing variable but it didn't work out right, i dont ...
1
vote
1answer
48 views
How to find the unknown values in this Numerical Integration type?
Given the following type of numerical integration:
$$I(f)=\int_0^1 f(x) \, dx \approx \frac 12 f(x_{0}) +c_1 f(x_1) $$
a) Find the values of: the coefficient $c_1$ and points $x_0$ and $x_1$ so ...
0
votes
1answer
33 views
Riemann integral show $f(x)=g(x)$ for at least 1 $x$ in [a,b]
Let $f$ and $g$ be continuous functions on $[a,b]$ such that $\int_a^b f = \int_a^b g$. Show that there exists $x\in [a,b]$ such that $f(x) = g(x) $.
I want to assume not and then show that the ...
1
vote
1answer
34 views
Monotonic integral proof
Let $f$ be a continuous function on $[a,b]$ such that $f(x) \geq 0 $
for every $x\in [a,b]$. Suppose $\int_a^b f = 0$ and show that $f (x) = 0$ for every $x\in [a,b]$.
obv this is monotonic ( ...
3
votes
0answers
30 views
Stokes' Theorem and Measure Zero Sets
This is probably a very naive question but I am trying to connect two pieces of information in my head regarding integration of differential forms and integration with respect to a measure.
The first ...
4
votes
3answers
89 views
Integral of rational functions.
I want to evaluate this integral:
$$\int{\frac{ax+b}{(x^2+2px+q)^n}}dx$$
The book only says to integrate by parts $\int{\dfrac{1}{(x^2+2px+q)^{n-1}}dx}$,
for simplicity if $n = 2$ I get:
...
1
vote
1answer
38 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
12
votes
4answers
278 views
Which methods to use to integrate $\int{\frac{x^4 + 1}{x^2 +1}}\, dx$
I have this integral to evaluate:
$$\int{\frac{x^4 + 1}{x^2 +1}}\, dx$$
I have tried substitution, trig identity and integration by parts but I'm just going round in circles.
Can anyone explain ...
2
votes
3answers
45 views
Find the antiderivative of $\sqrt{3x-1} dx$
Find the antiderivative of $\sqrt{3x-1} dx$.
I got $\frac{2}{3}(3x-1)^{3/2}+c$ but my book is saying $\frac{2}{9}(3x-1)^{3/2}+c$
Can some one please tell me where the $2/9$ comes from?
1
vote
1answer
29 views
Question about integrability
Let f be a continious function on [a,b] and exist a partition P of [a,b] such that $\bar{S}(f,p)=\int_a^b f(x)dx$. Prove that f is a constant function. I thought stratting assuming the claim is not ...
1
vote
2answers
60 views
Evaluate $\int \dfrac{1}{\sqrt{1-x}}\,dx$
Find $$\int \dfrac{1}{\sqrt{1-x}}\,dx$$
I did this and got $\dfrac23(1-x)^{\frac32} + c$
But a online calculator is telling me it should be $-2(1-x)^{\frac12}$
What one is on the money and if not ...
0
votes
1answer
14 views
expectation of logarithm under generalised inverse gaussian
I want to follow the following integral:
$$\frac{1}{C}\int_0^\infty \log(z)\,z^{p-1}\exp\left(-\frac{az+b/z}{2}\right)\,dz$$
where C is the normalising constant.
The following might be useful ...
1
vote
3answers
60 views
Is this an exact differential or not?
I have the 1-form
$$dz=2xy\, dx+(x^{2}+2y)\, dy$$
And I want to integrate it from $(x_{1},y_{1})$, to $(x_{2},y_{2})$.
If I'm not drunk, checking mixed partials, I find that $dz$ is an exact ...
0
votes
1answer
30 views
The sum of the integration of g and $g^{-1}$
Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto
$[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function.
Use geometric insight to visualize the ...
0
votes
3answers
63 views
How to solve these?
Inverse Trigonometric Functions
They are incomplete and I don't know how to complete them.
Who can help me?
1st
$$
\int\frac 1{ x \sqrt{x^{6} - 4}}dx
$$
I tried with:
$$u = x^3 $$
$$du= 3x^2dx$$
...
1
vote
1answer
48 views
Can someone demonstrate this integral with a cartesian product?
From this question, we can apparently get an integral:
$$\int_{a \times c}^{b \times d}\!{\left(1+e^{i(x+1/2y)}+e^{i(y)}\right)\,d(x \times y)}$$
...I'm not exactly sure that this integral is posed ...
5
votes
3answers
115 views
integration by substitution, using $\;t = \tan \left(\frac 12 x\right)$
$\displaystyle\int_0^\frac{\pi}{2}\frac{1}{2-\cos x} \, dx$ using the substitution $t=\tan\frac{1}{2}x$
$x=2\tan^{-1}t$
$\dfrac{dx}{dt}=\dfrac{2}{1+t^2}$
$dx=\dfrac{2}{1+t^2}\,dt$
...
1
vote
0answers
35 views
Product of Fourier integrals
I am interested in solving the following integral:
\begin{equation}
I =\int dx_{3}\psi^{\star}(x_{3})\int dx_{1}\psi(x_{1})\int dq_{1}X(q_{1})e^{iq_{1}(x_{3}-x_{1})}\int dx_{2}\psi(x_{2})\int ...
2
votes
3answers
61 views
Help evaluating $\int_0^\infty \frac{1}{x^{1/2}(x+1)}dx$
I began solving this with U sub and partial fractions...first for $x^{1/2}$ and then for $x+1$ but neither of those methods got me the answer of $\pi$.
I know the indefinite integral should be ...
2
votes
2answers
66 views
When $\int |f|=\left|\int f\right|$ holds?
I was just wondering when did the equality hold for the following inequality:
$$\left|\int_{R^d}f(x)\, d x\right|\leq\int_{R^d}|f(x)|\, d x$$
where $f:R^d\to R$ is Lebesgue integrable on $R^d$.
...
6
votes
1answer
105 views
Why substitution method does not work for $\int (x-\frac{1}{2x} )^2\, \mathrm dx$?
Why $$\int \ \left(x-\frac{1}{2x} \right)^2 \, \mathrm dx$$ is easy to integrate once $$\left(x-\frac{1}{2x} \right)^2$$ is expanded, but impossible using substitution method? (tried 5 different subs ...
4
votes
1answer
73 views
How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?
I have two relations:
1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$.
2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$.
From these two how does it follow that ...
0
votes
2answers
45 views
Integration by parts disconnect
I'm trying to integrate $\displaystyle E(Y^2) = \int^\infty_0 y^2\lambda e^{-\lambda y} dy$
doing it by parts this is my logic.
$\displaystyle E(Y^2) = \int^\infty_0 y^2\lambda e^{-\lambda y} dy$ ...
0
votes
0answers
38 views
Riemann integration show if f is integrable then g is integrable
I have the following question asked of me.
Suppose that $f$ and $g$ are bounded functions on $[a,b]$
and there exists a point $c\in[a,b]$ such that $f(x) = g(x)$ for every $x\neq c$.
Prove that $U(f) ...
3
votes
3answers
37 views
Integration of function help
I'm having problems integrating this function $\displaystyle E(X)=\int^ \infty_0 x\lambda e^{-\lambda x} dx$. I did the integration by parts and had $-xe^{-\lambda x}- \lambda e^{-\lambda x}$. However ...
4
votes
1answer
66 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
2
votes
1answer
22 views
Sequence of continuous functions, integral, series convergence
Let $f_k$ be a sequence of continuous functions on $[0,1]$ such that $\int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx$ for all $n \in \mathbb{N}$.
Is $\sum _{k=1} ^{\infty}f_k(x)$ convergent?
Could ...
0
votes
4answers
54 views
Integration problems
Can anyone help me with these:-
(a)Prove by induction: $\displaystyle\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
(b) By explicitly calculating upper and lower Riemann sums on a uniform partition and
...
1
vote
1answer
64 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
-7
votes
0answers
68 views
Why Riemann integration is needed? [closed]
What is the necessity of the notion of "Riemann Integration" ? Why is normal definite integral is not good enough ?
4
votes
2answers
44 views
Find the antiderivatives
Find the antiderivatives:
$\int\!\left(2x^2 +3\right)^{1/3} x\,dx $
I have hit this in my book and the way I do it I get
$3/4\left(2x^2 + 3\right)^{4/3} x^2 +c $
But my book tells me it should be ...
1
vote
1answer
49 views
Integration $\int \left(x-\frac{1}{2x} \right)^2\,dx $
$$\int\!\left(x-\frac{1}{2x} \right)^2\,dx $$
From U-substitution, I got $u=x-\frac{1}{2x},\quad \dfrac{du}{dx} =1+ \frac{1}{2x^2}$ , and $dx= 1+2x^2 du$
and in the end I come up with the answer to ...
3
votes
3answers
56 views
Integrate $\int {{{\left( {\cot x - \tan x} \right)}^2}dx} $
$\eqalign{
& \int {{{\left( {\cot x - \tan x} \right)}^2}dx} \cr
& = {\int {\left( {{{\cos x} \over {\sin x}} - {{\sin x} \over {\cos x}}} \right)} ^2}dx \cr
& = {\int {\left( ...
0
votes
1answer
63 views
How to integrate e to the power to the power?
How should I integrate?
$\int_0^\infty e^{-x^{1/3}}dx$
I think this is a simple question for the experts. But a bit hard to tell Google what I want. So, thanks for your help! :)
And this looks ...
1
vote
0answers
28 views
Upper and lower integration inequality
I would like to learn how to prove that the following inequality holds.
Let $F$ be a bounded function on an interval $[a,b]$, so that there exists $B\geq 0$ such that $|f(x)| \leq B$ for every $x\in ...
5
votes
3answers
86 views
Integral of $\cot^2 x$?
How do you find $\int \cot^2 x \, dx$? Please keep this at a calc AB level. Thanks!
5
votes
4answers
72 views
Integrate ${\sec 4x}$
How do I go about doing this? I try doing it by parts, but it seems to work out wrong:
$\eqalign{
& \int {\sec 4xdx} \cr
& u = \sec 4x \cr
& {{du} \over {dx}} = 4\sec 4x\tan 4x ...
0
votes
1answer
30 views
Prove $\int_2^\infty{\frac{\ln(t)}{t^{3/2}}},\mathrm{d}t$ converges
Show, using a comparison test, that $\displaystyle \int_2^\infty{\frac{\log{t}}{t^{\frac32}}}\mathrm{d}t$ converges.
All the answers I've tried shows it diverges, taking $\log{t} \le t^{1/2}$ and ...
0
votes
0answers
26 views
New differintegral formula: how is it related to other differintegral formulas?
Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x)= \lim_{t\to x} ...
1
vote
2answers
63 views
Calculate the following integral
I was thinking about the following problem:
Calculate $\displaystyle \oint_{C}(\bar z)^2 \mathrm{dz}$ where $C:|z-1|=1$ is oriented counter clock-wise.
My Attempt: I take $z-1=e^{i\theta}$ ...
