Questions on the evaluation of definite and indefinite integrals
0
votes
2answers
28 views
Curve defined by a vector
http://i.stack.imgur.com/tD4Bn.png
I'm studying line integrals with a curve as a vector, but I couldn't understand the 'dr' part.
First of all: the curve isn't really a curve, it's like some points ...
0
votes
1answer
34 views
Calculate the limit using definite integrals: $\lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2}$
Calculate the limit using definite integrals: $\lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2}$
Well, I started like this:
...
5
votes
2answers
58 views
Trigonometric function integration: $\int_0^{\pi/2}\frac{dx}{(a^2\cos^2x+b^2 \sin^2x)^2}$
How to integrate $$\int_0^{\pi/2}\dfrac{dx}{(a^2\cos^2x+b^2 \sin^2x)^2}$$
What's the approach to it?
Being a high school student , I don't know things like counter integration.(Atleast not taught ...
3
votes
3answers
110 views
What is $\;\int xe^{-x^2} \,dx\;?$
What is $$\int xe^{-x^2} dx\quad?$$
I used substitution to rewrite it as $$\int -\dfrac{1}{2}e^u\, du$$ but this is too hard for me to evaluate. When I used wolfram alpha for $\int e^{-x^2} dx$ I got ...
5
votes
0answers
36 views
Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $
While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have
\begin{align*}
&\int_{0}^{\infty} \left[ \left( ...
3
votes
5answers
79 views
Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$.
Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$.
It seems to me like a Riemann integral definition:
...
0
votes
0answers
24 views
Integrating: $+ du$ and $ \cdot du$
Let's consider the following examples:
Evaluate the integral of $6x \sqrt{x^2+1}$
Evaluate the integral of $3x^2 + \sin(x^3-1)$
If I use the substitution method, the first integral gives me the ...
1
vote
3answers
75 views
Is there a smarter way to integrate?
My textbook uses the following technique to integrate. Let us take the following as an example, evaluate the integral of:
$$f(x) = 6x(x^2 + 1)^5$$
Notice that $[x^2+1]' = 2x$, so $6x\space dx = 3 ...
5
votes
2answers
96 views
How to prove a generalized integral identity
$$
\int_{0}^{\infty }\frac{t}{(e^{2\pi t}-1)(1+t^{2})}dt=-\frac{1}{4}+\frac{\gamma}{2}
$$
where $\gamma$ = Euler Gamma
$$
\int_{0}^{\infty }\frac{t}{( e^{2\pi t}-1)(1+t^{2}) ^{2}}dt=\frac{\pi^2}{24} ...
2
votes
1answer
91 views
How to compute $\int_0^\infty \frac{\sin t}{t^{s+1}} dt $?
How to compute $\displaystyle\int_0^\infty \frac{\sin t}{t^{s+1}}\;\text dt$ ?
Here, the real part of the complex number $s$ is negative and greater than $-1$.
0
votes
0answers
42 views
Integrating along a line of points; two different approaches, two different problems
I have an integration question, and I've taken two different approaches which both have serious flaws and don't agree; I'll outline them here and would be grateful for any guidance. I have an infinite ...
1
vote
1answer
32 views
determine whether an integral is positive
Given a standardized normal variable $X\sim N\left(0,1\right)$, and constants $ \kappa \in \left[0,1\right)$ and $\tau \in \mathbb{R}$, I want to sign the following expression:
\begin{equation}
...
1
vote
0answers
38 views
What is the value of $\int_{0}^{a} \int_{0}^{x} \frac{x}{y} \cosh{y} \; dy \, dx$?
I had the following double integral in my recent math examination:
$$\int_{0}^{a} \int_{0}^{x} \frac{x}{y} \cosh{y} \; dy \, dx$$
where $$a \in \mathbb{R}$$
I tried changing the order of the ...
3
votes
1answer
65 views
Improper integral and special functions
I'd like to have an expression of the following integral:
$$\int_0^{+\infty} \left(\sqrt{1+x^4} - x^2\right) dx$$
in terms of some special functions (but not in the form given by Wolfram Alpha).
1
vote
4answers
106 views
How to compute the indefinite integral $\int \frac u{u+1}\,\mathrm du$?
How do you compute $$ \int \frac{u}{u+1}\,\mathrm du$$
1
vote
1answer
36 views
Integral involving normal densities
I am trying to solve the integral
$$I(y)=\int_{\mathbb R}f(x,y)g(x)dx,$$
where $f(x,y)$ is the bivariate normal density with known mean $(\mu_1,\mu_2)$ and covariance matrix $\Sigma$ , and $g(x)$ is ...
1
vote
1answer
52 views
Integral sign with circle (AND arrow on the circle) through it
I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path.
So when I encountered in complex analysis the above integral sign but with ...
0
votes
2answers
57 views
$\int_0^\infty |2 \arctan\sqrt{x} - \pi {x^p}/(1+x^p)| \mathrm dx $
Let $a(x) = 2 \arctan\sqrt{x}$ and $b(x) = \pi {x^p}/(1+x^p)$. I'm trying to evaluate:
$$\int_0^\infty |a(x) - b(x)| \mathrm dx $$
I've figured out I need the integrals:
$$\int_0^1 a(x)-b(x) dx = ...
1
vote
1answer
25 views
Find flow of the vector field $\overrightarrow{\operatorname{rot}F}$
We've given 5 points in $\mathbb{R}^3$: $A=(1,0,0)$, $B=(0,1,0)$, $C=(0,0,1)$, $D=(1,1,0)$, $E= (1,1,1)$. We have a surface $S$ given by triangles $ADE, DBE, BCE, CAE$. We have a vector field: ...
0
votes
1answer
45 views
How to find the primitive function of this integral
I am trying to find the primitive function of $\displaystyle\int_{}^{}\frac{dx}{5+2\sin x-\cos x}$. I've got $$\int_{}^{}\frac{dx}{5+2\sin x-\cos ...
0
votes
2answers
41 views
Vector Line Integral Question
I need to compute the line integral for the vector $\vec{F} = \langle x^2,xy\rangle$, for the curve specified: part of circle $x^2+y^2=9$ with $x \le0,y \ge 0$,oriented clockwise.
Once again, I'm ...
5
votes
3answers
56 views
Applications of Double/Triple Integrals
This is the question that I need to solve using mathematica:
The concentration of an air pollutant at a point $(x,y,z)$ is given by: $$p(x,y,z) = x^2y^4z^3 \text{ particles}/m^3$$ We're interested ...
3
votes
3answers
122 views
Finding trigonometric integral (challenging)
Integrate:
$$\displaystyle\int \dfrac{(1+\cos^2A\times\cos(2A))^2}{\cos(2A)\times(2\cos^4A+\sin^2A)}dA$$
Again my working with trigonometric identities has not take me anywhere useful, that is ...
-2
votes
0answers
39 views
calculate volume in spherical coordinates
In calculate volume enclosed within the cone of $z^2=4(x^2+y^2)$ and inside sphere $x^2+y^2+(z-2)^2=4z-3$ in spherical coordinates.
what's limit of $\rho$?
2
votes
2answers
45 views
Calculate the length of curve $f(x)=\arcsin(e^x)$, check solution, please.
As in the topic, my task is to calculate the length of $f(x)=\arcsin(e^x)$ between $-1, 0$. My solution: I use the the fact, that the length of $f(x)$ is equal to $\int_{a}^b\sqrt{1+(f'(x))^2}dx$ ...
1
vote
0answers
30 views
Integral of Bessel function of the first kind and exponential function
I would need to know if there's a closed form for the following integral:
$$\int_{0}^{\infty} x^{-1}J_{\frac{1}{2}}(\pi x)J_{\frac{1}{2}}(\pi x)\exp(-b(x-x_0)^2)$$
with $b>0$ and $x_0\in ...
1
vote
0answers
36 views
Basic Trigonometric Substitution Question
I have a basic trig substitution question for integrals. It always seems that x is opposite to the theta angle. However, making x the adjacent on the right angle triangle seems to work just fine as ...
0
votes
0answers
21 views
$\mathrm{E}[\log (1 + a X)]$ for non-central chi-squared distributed $X$
I'm really not great in analysis, so I recently got stuck on this problem (Please correct me if I'm wrong somewhere):
Let $Y \sim \mathcal{N}(\mu, 1)$ be a random variable with mean $\mu$ and normal ...
2
votes
2answers
67 views
How to calculate $\lim\limits_{x\to1^+}\frac{1}{(x-1)^2} \int\limits_{1}^{x} \sqrt{1+\cos(\pi t)}\,\mathrm dt$
Can anyone help me by calculating this limit?
I know that I need L'Hôpital but how?
$$
\lim_{x \to 1^+}\frac{1}{(x-1)^2} \int_{1}^{x} \sqrt{1+\cos(\pi t)} \,\mathrm dt
$$
Thank you very much!!
15
votes
4answers
1k views
Why doesn't integrating the area of the square give the volume of the cube?
I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):
$$V = \int\limits_a^b {A(x)dx}$$
But this doesn't seem to work with the square. ...
0
votes
2answers
139 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
7
votes
4answers
177 views
How prove this $\int_{0}^{\infty}\sin{x}\sin{\sqrt{x}}dx=\dfrac{\sqrt{\pi}}{2}\sin{\left(\dfrac{3\pi-1}{4}\right)}$
prove that
$$\int_{0}^{\infty}\sin{x}\sin{\sqrt{x}}dx=\dfrac{\sqrt{\pi}}{2}\sin{\left(\dfrac{3\pi-1}{4}\right)}$$
I have some question,use the ...
1
vote
1answer
49 views
How to calculate this multi-integral?
Please calculate $$I=\int_0^1 dx \int_0^x dy \int_0^y \frac{\sin z}{(1-z)^2}dz$$
Any hints? Thank you!
10
votes
1answer
128 views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
2
votes
1answer
42 views
Closed Form of a Particular Sum
Does anyone have any ideas on how to find a closed form for the following expression? It comes up when trying to bound a particular integral. The sum is:
$$\sum_{n=0}^{\infty} ...
0
votes
1answer
34 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
3answers
42 views
Trigonometric substitution integral
Trying to work around this with trig substitution, but end up with messy powers on sines and cosines... It should be simple use of trigonometric properties, but I seem to be tripping somewhere.
...
1
vote
0answers
34 views
Integral of gaussian and sine/cosine
I really need the solution of two integrals involving exponentials and sine/cosine.
For $n\in \mathbf{N}$ even :
$$\int_{-\infty}^{+\infty}\left(\frac{2\,ax\sin(\pi ...
4
votes
3answers
90 views
Integral of rational functions.
I want to evaluate this integral:
$$\int{\frac{ax+b}{(x^2+2px+q)^n}}dx$$
The book only says to integrate by parts $\int{\dfrac{1}{(x^2+2px+q)^{n-1}}dx}$,
for simplicity if $n = 2$ I get:
...
1
vote
1answer
41 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
1
vote
1answer
40 views
Apparently simple integral
I am having trouble solving this apparently simple integral:
$\int\frac{x}{3+\sqrt{x}}dx$
Hints would be preferable than complete answer...
Thanks!
13
votes
3answers
142 views
Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$
I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
12
votes
4answers
281 views
Which methods to use to integrate $\int{\frac{x^4 + 1}{x^2 +1}}\, dx$
I have this integral to evaluate:
$$\int{\frac{x^4 + 1}{x^2 +1}}\, dx$$
I have tried substitution, trig identity and integration by parts but I'm just going round in circles.
Can anyone explain ...
1
vote
0answers
19 views
Closed form for $k$-th moment
I would like to calculate this $k$-th moment:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(i^n\frac{\sin(\pi a x+\frac{n\pi}{2})}{\pi ax+\frac{n\pi}{2}}+(-i)^n\frac{\sin(\pi a ...
1
vote
0answers
14 views
simplification of a complex expression
I am collecting proofs for certain integrals. To simplify certain proofs, I use
$e^{Ax}cos(Bx)=\mathcal{Re}[e^{(A+iB)x}]$, where $A$, $B$, and $x$ are real. Is there an analogous simplification for $ ...
0
votes
1answer
25 views
An integral identity, general multidimensional case
A follow-up to this question. Which of the following equations is true? Can someone give a proof to the correct equation. Thanks.
Please note the prefactors $N$ and $N!$
$$\int_0^A dx_1 \ldots ...
0
votes
1answer
19 views
Show relation for integrals
Let $f \in C^{1}([a,b];\mathbb{R})$ and $|f'(x)-f'(y)| \le L |x-y|$
then we have $|\int_a^b f(x) dx -f(\frac{a+b}{2})(b-a)| \le L\frac{(b-a)^3}{4}$.
I have troubles to show this inequality. the ...
0
votes
1answer
13 views
Body Volume Rotation Of Shape Question
I want to know if I`m following the correct step to evaluate the body volume rotation of shape.
my function is : $$y=ln(x)$$ and I want to evaluate the body volume rotation of it between
$y=0$ and ...
0
votes
0answers
21 views
Question About Indefinite Integrals
I`m trying to understand how should I evaluate this indefinite integral with this data on the integral :
the question is : "Draw the shapes on the plain blocked - by the data lines and evaluate"
:
1) ...
0
votes
1answer
34 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 ...
