All Questions
111
questions
3
votes
1
answer
115
views
Computing an integral using differential under the integral sign
The following integral is in question.
$$I(x) =\int_0^x \frac{\ln(1+tx)}{1+t^2}\,dt$$
My attempt is finding $I’(x)$ which is
$$I’(x) = \int_0^x \frac{t}{(1+t^2)(1+tx)}\,dt + \frac{\ln(1+x^2)}{1+x^2} $$...
- 617
0
votes
1
answer
62
views
If $y^2 - (x^2)y -2x=0, x, y >0$ and $\int \frac{(y-x^2)}{(x^2 +y)(y^2 +x)} dx=f(y) +C$ , where $c$ is a constant, find the value of $f'(y)$ at $x=1$. [closed]
I need help with this question:what I have done so far
$y^2 - (x^2)y -2x=0 \\
y-x^2 =\frac{2x}{y}...(1)$.
Substituting $1$ in integration I get $\int \frac{(2x)}{y(x^2 +y)(y^2 +x)} dx$. After this I ...
1
vote
0
answers
49
views
Differentiation for vector-valued functions is interesting. Integration for vector-valued functions is uninteresting. Why?
I am reading "Introduction to Analysis 1" by Mitsuo Sugiura (in Japanese).
Mitsuo Sugiura defined differentiation for vector-valued functions in the book.
The following definition from "...
- 1,138
0
votes
1
answer
51
views
Steepest Ascent Theorem
The temperature on the $Oxy$ plane is given by
f$(x, y) = 100−x^2 −2y^2$.
Assuming that an ant follows the path of steepest ascent of temperature, find and sketch
the path the ant follows if it starts ...
- 261
0
votes
0
answers
33
views
Time derivative of integrals on changing level and upper contour sets
There are related questions and answers on StackExchange but I cannot find a general formula for the following types of (time) derivatives of integrals with respect to changing level and upper contour ...
- 93
5
votes
1
answer
131
views
How to find the speed to maximise the distance travelled
I'm given that a car travels at speed along its track it uses fuel at a rate of
$R(v)=17v^2 –7v+7$, in litres per sec.
Where $v$ is the speed in metres per second.
The fuel tank holds $Y$ litres of ...
0
votes
1
answer
60
views
Indefinite integral functions with vector inputs
I would sincerely appreciate your help on an integral-related question.
I have two real-valued functions $x(t): \mathbb{R} \rightarrow \mathbb{R}$ and $y(t): \mathbb{R} \rightarrow \mathbb{R}$, and ...
- 25
0
votes
2
answers
209
views
Solving this probability function
This is the probability of winning a game I calculated. It stands for the probability that playerX wins given he has a limit of x, and playerY has a limit of Y (the games involves picking a limit). If ...
user1156732
0
votes
1
answer
44
views
Why is $\int H(t,c(t),c^\prime(t)) dt \in \mathcal{C}^r$ if $H \in \mathcal{C}^r$?
In lecture came up the following remark:
Consider an interval $I = [t_0,t_1]$ and a finite dimensional Banach space $X$. Let $U$ be an open subset of $\mathbb{R} \times X \times X$ and let $V \...
- 4,222
2
votes
1
answer
203
views
integration with respect to time
I am pretty sure this is a really basic question, but after the summer, not using integrals / derivatives at all, I just can not remember how to do math anymore.
I have a function for acceleration ...
- 23
1
vote
1
answer
65
views
Derivative through the integral of two variables function
Problem: Let $h$ be a function in $C^1(\mathbb{R})$. Calculate the following derivative
$$\dfrac{d}{dt}\int_{0}^{t}h(s,x-c(t-s))ds.$$
My first attempt: I have searched for a way to solve it and find ...
- 1,238
0
votes
1
answer
136
views
Deriving the potential function for a gradient field
I am trying to write out the general formula for finding the potential function from a gradient field; but I'm encountering an issue.
My current working is:
$$Let\space F = f(x,y,z)$$
$$\nabla\vec{F} =...
- 3
2
votes
1
answer
75
views
Special case of multi variable integration with natural log result?
We know the famous equation $PV = nRT$ but in thermodynamics we typically deal with differentials $P\,dV$ and even $V\,dP$ at times.
Given:
$$
d(PV) = V\,dP + P\,dV
$$
integrating both sides
$$
PV = \...
- 131
0
votes
0
answers
67
views
Integral of arctan of two functions
I'm dealing with an integral of the form
$$
I = \int_{x_1}^{x_2} \frac{f_1 \, \text{d}f_2 \, - f_2 \, \text{d}f_1}{f_1^2+f_2^2}
$$
where $f_1:=f_1(x)$ and $f_2:=f_2(x)$ and there are no failures of ...
- 79
0
votes
1
answer
33
views
partial derivatives under integral sign
We have $f(x,y)=\int_{0}^{y^2\sqrt{x}}\sin(t^2)dt$. How does one prove that $f$ is differentiable on $0<x<\infty,-\infty<y<\infty?$ And how does one calculate $\frac{\partial{f}}{\partial{...
