All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
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,
$$\mathbb{D}^s_xf(x)= \lim_{t\to x} ...
1
vote
2answers
63 views
Calculate the following integral
I was thinking about the following problem:
Calculate $\displaystyle \oint_{C}(\bar z)^2 \mathrm{dz}$ where $C:|z-1|=1$ is oriented counter clock-wise.
My Attempt: I take $z-1=e^{i\theta}$ ...
1
vote
2answers
76 views
Find the area of $A = \{ \langle x,y\rangle \in \mathbb{R}^2 \mathrel| (x+y)^4<a x^2 y,\ x>0 \}$?
I can't really think of how to set the limits
3
votes
3answers
70 views
Evaluate $\int x \cos x^2 dx$
Hay I have hit this in my book
Evaluate $\int x \cos x^2 dx$.
I got $x^2 \sin(x^2) / 2 $
But I used a online calculator to check it and it is giving me $\sin(x^2)/2 $
Where dose my X go?
2
votes
0answers
34 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
3
votes
4answers
107 views
Integrate by parts: $\int \ln (2x + 1) \, dx$
$$\eqalign{
& \int \ln (2x + 1) \, dx \cr
& u = \ln (2x + 1) \cr
& v = x \cr
& {du \over dx} = {2 \over 2x + 1} \cr
& {dv \over dx} = 1 \cr
& \int \ln (2x ...
0
votes
2answers
49 views
An integral problem?
How do you integrate $e^{e^x}$? I was able to get it down to du/(ln u) but I wasn't able to go further. Thanks!
2
votes
1answer
53 views
Simplification of an expression
How do I simplify the following expression?
$$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$
where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , $e ...
0
votes
0answers
44 views
separating a variable from integral
In the following integral, I would like to separate $\alpha$ from rest of the equation. Can we solve the following equation for $\alpha$?
$$\large{\int_{0}^{a} \int_{0}^{2\pi} ...
1
vote
1answer
41 views
$\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$
$$\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$$
Please help me in this integral, I've tried some substitutions, but nothing work.
Thanks in advance!
-1
votes
1answer
67 views
$ \underset{r \rightarrow 0+}{\lim} \int ^1 _r x^{2013}(\ln x)^{1001} dx$
This integral is in my book, but I don't know how to solve it using simply methods. It's possible to do it clever.
$$ \underset{r \rightarrow 0+}{\lim} \int ^1 _r x^{2013}(\ln x)^{1001} dx$$
Could ...
1
vote
2answers
47 views
Evaluating the integral: $\lim_{R \to \infty} \int_0^R \frac{dx}{x^2+x+2}$
Please help me in this integral:
$$\lim_{R \to \infty} \int_0^R \frac{dx}{x^2+x+2}$$
I've tried as usually, but it seems tricky. Do You have an idea?
Thanks in advance!
1
vote
2answers
48 views
Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution
Consider this integral:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$$
How would you compute it?
I already solved this problem this way:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \left( ...
3
votes
2answers
84 views
$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$ , $\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$
Please help me integrate
$$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$$
and
$$\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$$
I've tried the standard $u = \tan \frac{x}{2}$ substitution but it looks ...
0
votes
1answer
34 views
Show that E is measurable?
Suppose $E_1= [1, 1 \frac12] , E_2 = (2, 2\frac14), E_3 = [3, 3\frac18], E_4 = (4 , 4 \frac{1}{16}) , \dots , E= \bigcup_{n=1}^{\infty}E_n
$
i) Show $E$ is measurable
ii) Compute $m(E)$
Here is ...
4
votes
3answers
90 views
Integrating left to right versus right to left.
OK, I understand that when integration is done left to right with respect to x increasing left to right (dx is positive), that the answer is positive, and vice versa when integrating right to left. ...
2
votes
0answers
39 views
Interchanging the limiting operations
How to remember the conditions for interchanging the limiting operations , for example between limits and integrals or integrals and sums or derivation of any order and integrals, i mean every one of ...
3
votes
5answers
131 views
3
votes
0answers
30 views
What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?
I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$.
...
1
vote
3answers
90 views
$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?
Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities:
$$\|f*g\|_q\leq ...
1
vote
1answer
36 views
Simple Diffy-Q problem
So as a fun project, I'm trying to work my way through Kreyzig's "Advanced Engineering Mathematics". But I've gotten to a really simple problem:
$$xy' = 2y$$
where I know the solution is $x^2$ but ...
0
votes
2answers
35 views
How to solve this integral $\int \frac{(1+2x^2)}{x^2(1+x^2)}dx$
Problem : How to solve this integral
$\int \frac{(1+2x^2)}{x^2(1+x^2)}dx$
I thought it should be $ x + 3x^2$ in the numerator so that I will take $x+x^3$ = u then taking derivative both sides and ...
1
vote
0answers
22 views
help on antiderivative of a vector function
Suppose I have $f(x) = {Sx\over ||x||}$, where $S$ is a $n\times n$ symmetric matrix and $x$ is a vector of length $n$. I'd like to know the antiderivative of $f(x)$, which is a real-valued function ...
2
votes
1answer
51 views
Riemann integral with intervals?
Let $f(x) = \begin{cases} 3 && 0 \leq x \leq 1 \\ 0 && 1 \leq x \leq 2 \end{cases}$
Compute $\displaystyle \ \ \int_0^2 f(x)dx\,\,\,$.
You can use the definition of Riemann integral ...
0
votes
2answers
58 views
Integrating $\int{\frac{1}{1+e^{x}}}dx$, Partial Fractions(?)
I need help with this integral:
$$H(x) = \int{\frac{1}{1+e^{x}}}dx$$
It should be easy, but I'm stuck. I thought about using a u-substitution but I didn't get any further. Am I meant to use partial ...
1
vote
3answers
55 views
Using Spherical coordinates find the volume:
Inside the surfaces $z=x^2+y^2$ and $z=\sqrt{2-x^2-y^2}$
I integrated over the ranges:
$0 \leq \theta \leq 2\pi$
$ 0 \leq \phi \leq \frac{\pi}{2}$
$0 \leq r \leq \sqrt{2}$
I get ...
2
votes
1answer
40 views
How to prove Chebyshev–Gauss quadrature integrate polynomial of degree less than $2n-1$ exactly
What I want to ask is mentioned in the title.
For example: how can we show that ...
1
vote
0answers
21 views
Gaussian quadrature with arbitrary weight function
In class, our professor told us how to evaluate the integral $\int_a^bw(x)f(x) dx$ by finding the Gaussian nodes $x_i$ and weight $w_i$ with weight function $w(x)=1$ (also known as Legendre ...
2
votes
3answers
41 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
0
votes
4answers
91 views
How to integrate $\int_0^\infty e^{-ty^2} \sin t dt$
My book suggests that I do some sort of limiting
$\lim_{A \to \infty} \int_0^A e^{-ty^2} \sin t d t$
But I'm not getting anywhere.
1
vote
2answers
41 views
Primitive function of $x^3 \sin x^2$
I'm trying to find the primitive function of $x^3 \sin x^2$, and I've come to a variable exchange ($t = x^2$) which led me to $\frac{1}{2} \int t \sin t dt$.
According to my text book, the primitive ...
3
votes
3answers
102 views
How do you integrate the following trigonometric function involving sin and cos?
How do you integrate the following functions:
$$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta
$$
...
0
votes
1answer
54 views
Theorem or just a change of varibles?
I have a formula in my text:
$$\int \int_{S} F \cdot n dA= \int \int_{w} F(G(u,v)) \cdot (dG_{u}\times dG_{v}) du dv$$
I am really lazy and hate remembering formulas to me this looks like a ...
2
votes
1answer
46 views
When is it valid for me to just integrate a trig function?
I am having a problem identifying when I need to use some kind of integration technique or am I just over complicating things. Could someone please explain to me when I need to or not?
Normally, I ...
2
votes
1answer
30 views
Evaluating a Gaussian Integral
How to prove that
$$\int_{\mathbb{R}^N}e^{-\langle Ax,x\rangle}\operatorname{dm}(x)=\left(\frac{\pi^N}{\det A}\right)^{\frac{1}{2}}$$
Where $A:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is a symmetric ...
1
vote
0answers
50 views
Integral involving Gauss Hypergeometric function, power, exponential and Bessel Function
I am trying to evaluate the following integral involving the Gauss Hypergeometric function, power, exponential and a Bessel Function:
$$
\int_0^\infty x e^{-cx^2} {_2F_1(1,\frac{2} {ab},1+\frac{2} ...
0
votes
0answers
59 views
Integral involving exponential, power and Bessel function
Is there any formula for calculating the following definite integral, including exponential and Bessel function?
$$ \int_0^{a}x^{-1} e^{x}I_2(bx)dx$$
Thanks in advance
2
votes
0answers
63 views
A photon in expanding Universe (a snail on a tree)
I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
0
votes
0answers
20 views
Slightly tricky integration (proving B-spline property)
Given is that $N_1(x) = \chi_{[0,1]}(x), x \in \mathbb{R}$ and $N_{m+1}(x) = (N_m * N_1)(x), x \in \mathbb{R}$ assuming we have defined $N_m$. So we have
$$N_{m+1}(x) = \int_{-\infty}^{\infty} ...
4
votes
4answers
103 views
Definite Integral of square root of polynomial
I need to learn how to find the definite integral of the square root of a polynomial such as:
$$\sqrt{36x + 1}$$ or $$\sqrt{2x^2 + 3x + 7} $$
EDIT: It's not guaranteed to be of the same form. ...
-2
votes
1answer
46 views
Integrate the function $f(x, y, z)=\sqrt{3x^2 + 3y^2 + z + 1}$ over the surface given by…
Integrate the function $f(x,y,z) = \sqrt{3x^2 + 3y^2 + z + 1}$ over the surface given by the graph of $z = g(x,y) = x^2 + y^2$ over the region $1 \leq x^2 + y^2 \leq 4$
3
votes
2answers
51 views
Using area=$\int_{y = a}^{y = b} x \, dy $ find the following shaded area:
So I did:
$$\eqalign{
& y = {1 \over x^2} \cr
& {x^2} = {1 \over y} \cr
& x = y^{ - {1 \over 2}} \cr
& \int_a^b x \, dy = \left[ 2{y^{1 \over 2}} \right]_{1 \over ...
