Tagged Questions
2
votes
1answer
80 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
5answers
94 views
How do we differentiate an integral with a different argument
Let's say we have:
$$y = \int_0^x e^{t^2} dt$$
and we want $dy/dx$.
Do we consider $t$ as being $x$?
2
votes
2answers
55 views
$\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx$
Prove that: $\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx=\int_{-\infty}^{\infty} -2x e^{-x^2} \sin(2tx) dx$
This is my proof:
$\forall \ t \in \mathbb{R}$ (the improper integral ...
1
vote
1answer
91 views
Can the integral of $x^x$ be found?
I'm interested in knowing if the indefinite integral of $x^x$ can be found in terms of elementary functions.
I am under the impression (be it correct or incorrect) that it can be found. This is why: ...
2
votes
1answer
43 views
Can we use (higher order) derivatives to help integration?
I'm wondering if it's possible to use derivatives to ease the evaluation of an integral. For instance, I know that to evaluate an integral with enough precision I need to evaluate it at $n$ points. ...
1
vote
1answer
25 views
Related rates & derivatives/integration
A ladder 10 feet long leans against a vertical wall. If the bottom if the ladder slides away from the base of the wall at a speed of 2ft/sec, how fast is the angle between the ladder and the wall ...
1
vote
1answer
35 views
What are definite integrals that use functions in one or both of its limits?
I've seen these types of questions before, but I think I missed a formal explanation of them. I have a solution to the question in front of me, so my question is not related to what the answer is - ...
1
vote
4answers
51 views
Using integration and differentiation to solve for deceleration?
PROBLEM STATES:
The landing velocity of an airplane (at which it touches the ground) is 100mi/hr. It decelerates at a constant rate and comes to a stop after traveling .25miles along a straight ...
3
votes
1answer
46 views
Not Riemman integrable derivaitive
Is there a function $F:[a,b]\to\mathbb{R}$ differentiable with $F' = f$, but $f$ is not Riemann integrable on $[a,b]$. $[a,b]$ is a bounded interval?
Motivation
Rudin page 152 Theorem 7.17: Suppose ...
0
votes
0answers
44 views
Calculus with Exponential Matrix
I have a following with derivating and integrating using exponential Matrix. Kindly have a look at it.
Consider $A\in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}$, $u(t)\in ...
2
votes
2answers
104 views
Derivative of an integral
I would like some guidance on how to solve these type of problems.
Find $h'(x)$ if $$h(x) = \int\limits_{\cos(x)}^x \mathrm{e}^{t^2} \, dt$$
Mathematica says $h'(x) = e^{x^2} - e^{\cos^2(x)} ( - ...
0
votes
1answer
46 views
Proving an integral/derivative identity.
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ real numbers. I wanted to prove that for $g:\mathbb{R}\to\mathbb{R}$ defined as $g(x)=\int^{b}_{a}f(x+t)\,dt$, $g$ is ...
2
votes
1answer
203 views
Integration by Parts and Leibniz Rule for Differentiation under the Integral Sign
Basically a friend of mine and I have had this hot debate for a little too long, I contend that these two tools are not only logically unconnected but they require different assumptions (I believe one ...
2
votes
1answer
39 views
Uniqueness of derivative
I don't know how to phrase the title better.
We say that a function is "growing" (don't know the English term) if
$f'(x)\ge0, \forall x$
However, if we want to say that the function is "strictly ...
3
votes
1answer
100 views
Differentiation under the double integral sign
Working on three-body dispersion forces I got the following quantity:
$$\frac{\partial }
{{\partial \lambda }}\int\limits_\lambda ^{\pi - \lambda } {d\theta } \int\limits_\lambda ^{\pi - \lambda } ...
2
votes
3answers
90 views
Can we use integration to simulate a derivative?
If we suppose that we have the problem of finding the derivative for some function, say $f(x)$ at a point $p$, can we use an integral to calculate it?
My incomplete idea is that we can take an ...
0
votes
0answers
43 views
variational problem
I have: $\Omega \subset R$, be open and bounded, assume that $q \in L^{\infty}(\Omega)$ satisfies $q\geq 0$ a.e in $\Omega$, and let $f :\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such ...
2
votes
1answer
184 views
How does differentiation under the integral sign work?
From what I gather, it looks like you can use the method when your function depends on a variable and also a parameter. If you are given some definite integral that depended only on a variable, how ...
2
votes
1answer
159 views
Pursuit curves solution
For our math class we have to do some calculations with respect to pursuit curves. The chased object starts at point $(p,0)$. Chaser starts at $(0,0)$.(x,y) Speed of the chased object is $u$. Speed ...
1
vote
2answers
86 views
Particular solution of the inhomogeneous equation by using the method of undetermined coefficients
$$y_2^{\prime \prime} +y_2= -g_2p_1^2 \cos ^2 \tau + \omega_1p_1 \cos\tau $$
the differentiation with respect to time.
Solution of the homogeneous term = $A \cos\tau+ B \sin\tau$.
Now I want to ...
1
vote
3answers
89 views
Integration by parts question,, possibly a circular example [duplicate]
I am having trouble figuring this out.
$$\int_0^{1/3} \sec^3(\pi x) \, dx$$
We are currently doing integration by parts,, so I set $g(x)=\sec^3(\pi x)$ and $f'(x)=1$.
I arrived at: $$x\sec^3(\pi x) ...
1
vote
2answers
171 views
What is Leibnitz rule for a double integral?
My question is similar to this one here but a bit different.
I want to know the result of the following operation. I'm familiar with Leibnitz' rule but can't see how to extend it to this case. If it ...
-1
votes
1answer
147 views
Which method is efficient is compared to the other for numerical integration?
Out of the following methods for numerical integration which is one is best? I want to to know which of the following involves least amount of calculations. If someone can sort following in the order ...
1
vote
0answers
60 views
Derivatives and integrals, normal and fractional, and their explanations and relations
Assuming, naively, that one acquires the nth derivative of a function by repeatedly differentiating and finding a pattern. Thus one gets $f^{(n)}(x)=g(x,n)$. I have a few questions about this ...
0
votes
1answer
53 views
Where does the linearity of differential operators come from?
I just quoted the linearity of a differential operator, namely d/dz, in a proof and I was wondering where the root of this linearity lies. All of the differential operators which I have encountered ...
1
vote
1answer
22 views
Satisfying a Differential Equation
So, $y=2cos(kt)$
Therefore, $y'=-2sin(kt)k$
Thus, $y''=-2cos(kt)k^2$
Plugging this into $4y'' = -16y$ ...
$4(-2cos(kt)k^2)=-16(2cos(kt))$
I've simplified it this far:
$(cos(k)k^2+4cos(k))=0$
...
0
votes
2answers
43 views
How to find the equation? (Integral problem)
Here is a problem in our homework, I've never seen an integral written like that.. what does this question mean and how to solve it? Help please!! :(
Given that $\displaystyle ...
0
votes
1answer
47 views
inhomogeneous ordinary differential equation
How to solve the inhomogeneous ordinary differential equation
$$\ddot{\phi}_2 + \phi_2 + g_2\phi_1^2 + \omega_1\ddot{\phi}_1 = 0,$$
where
$$ \phi_1 = p_1 \cos(\tau + \alpha), $$
solution is given ...
3
votes
3answers
250 views
Discontinuous derivative.
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is ...
0
votes
4answers
142 views
prove , if $ \frac {dy}{dx} = 0 $ then $ y = c$ for some constant $c$ .
we all know that if $ y = c $ for some constant $c$ then , $ \frac{dy}{dx} = 0 $
but how can we prove the other way ? , i mean , how can we
prove that , if $ \frac {dy}{dx} = 0 $ then $ y = c$ for ...
7
votes
2answers
160 views
Definite integral $\int_0^{\frac{\pi}{2}}\! \frac{\sin x }{\sqrt{\sin 2x}}\,\mathrm{d} x$
$$\int_0^{\frac{\pi}{2}}\! \frac{\sin x }{\sqrt{\sin 2x}}\,\mathrm{d} x$$
I'm pretty sure I can finish it after finding the anti derivative. I tried changing the denominator to $2\sin x \cos x$ and ...
1
vote
2answers
104 views
What is this question asking? (integrals, derivatives)
The question is:
Let $G(x) = \int_{\cos(2x)}^{1/2} \arcsin(t) dt$. Find all $x \in [0,\pi/2]$ such that $\frac{d}{dx} G(x) = 0$.
Is it asking "for what values of x between 0 to $\pi/2$ is the ...
2
votes
2answers
138 views
Integral of $f(x) \exp(ikx)$ with finite bounds calculated using Fourier transform, and its derivative
I have an integral which I need to calculate numerically along the lines of
$$
I(k)=\int_0^{L} \exp(i k x)f(x) dx
$$
where $x$ and $L$ are real. $f(x)$ is not necessarily periodic and differentiable ...
2
votes
4answers
112 views
Show $\frac{d}{dx} \int\limits^{a(x)}_{0} f(x,y)dy = \int\limits^{a}_{0} \frac{\partial f}{\partial x}dy + a'(x)f(x,a)$
I am trying to show that
$$
\frac{d}{dx} \int\limits^{a(x)}_{0} f(x,y)dy = \int\limits^{a}_{0} \frac{\partial f}{\partial x}dy + a'(x)f(x,a)
$$
I know this has something to do with the fundamental ...
0
votes
0answers
80 views
Differentiation with respect to integral boundary
Using the chain rule show the following proposition:
Let $f$ be continuously on $[a,b]$ and $g:J\to[a,b]$ continuously differentiable for an interval $J$. We write
...
4
votes
1answer
71 views
Using Fundamental Theorem of Calculus “Twice”
Find $ F''(1)$ if
$$ F(x) = \int_1^x f(t)dt $$
$$ f(t) = \int_1^{2t} \sqrt{1+u^3} du $$
My work:
From the looks of it, it looks like the Fundamental Theorem of Calculus, twice.
From FTC... $F'(x) ...
0
votes
1answer
141 views
Finding Derivatives Using Fundamental Theorem of Calculus
Let's say that $$\ g(x) = \int_a^b (u^2-1)/(u^2+1) du$$ where $a=2x$ and $b=3x $ (Sorry, I couldn't figure out how to properly MathJax this)
The question asks to find the derivative using FTC. I had ...
0
votes
2answers
96 views
Derivative inside integral of a translated function
The exercise state as follows:
Let $f:R \rightarrow R$ be an absolutely continuous function over every compact in $R$.
I'm asked to prove that:
$\displaystyle \frac{d}{dy}\int_a^b f(x+y)dx=\int_a^b ...
0
votes
4answers
63 views
Taking the derivative of some function
Simple question about some function:
I define $$y =\exp\left(\int^x f(t)w(t) dt \right) $$ and i want to take the derivative with respect to x. Can I just say y'= $$f(x)w(x) \exp \left(\int^x f(t) ...
1
vote
2answers
83 views
Derivatives vs Integration
Given that the continuous function $f: \Bbb R \longrightarrow \Bbb R$ satisfies
$$\int_0^\pi f(x) ~dx = \pi,$$
Find the exact value of
$$\int_0^{\pi^{1/6}} x^5 f(x^6) ~dx.$$
Let
$$g(t) ...
1
vote
0answers
68 views
Evaluation of function: derivative,integral, absolute continuity
I'm in trouble with this exercise tough I thought it was not that difficult at the begininng, now I definitely changed my mind.
So I hope someone could help me to solve it. Give the function:
$f:[0,1] ...
0
votes
1answer
98 views
Derivative of a Parameter Integral (involving a Gaussian density)
I have a question about the derivative of a certain parameter integral. To fix notation, let $\phi_{\mu,\sigma}(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp{\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right)}$ denote ...
9
votes
2answers
390 views
Evaluating $\int_{0}^{x} e^t \sqrt{2 + \sin(2t)} \, dt$
I was recently asked to evaluate the following integral:
$$\int_0^x e^t \sqrt{2 + \sin(2t)} \, dt$$
It was beyond the ken of WolframAlpha, which I find quite discouraging.
Does anyone have an idea ...
1
vote
2answers
61 views
Integration property
I am having a problem with a question. Can somebody help me please.
Show that if $f(x)=\int_{0}^x f(t)dt$, then $f=0$
Thank you in advance
1
vote
0answers
28 views
variation of a final state due to changes in period (where the period is a parameter)
I have a simple ordinary differential equation
$\frac{dx}{dt}=f(x,t,p,T)$
$x(0) = x_0$, $x(T) = x_T$
where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks!
NOTE: I ...
2
votes
1answer
90 views
Integrating $\int \dfrac{2x \ln x}{\sqrt{x^2-9}}\mathrm dx$ by parts.
We have to integrate $$\int\frac{2x \ln x}{\sqrt{(x^2-9)}} \mathrm dx$$
Is it right to use Integration by Parts?
I tried to substitute it with
$$u = \log x,\: \mathrm du = \frac 1x \mathrm dx;$$
$$v ...
3
votes
2answers
142 views
Using LDCT to show a function is continuous and differentiable
We have the following test prep question, for a measure theory course:
$\forall s\geq 0$, define $$F(s)=\int_0^\infty \frac{\sin(x)}{x}e^{-sx}\ dx.$$
a) Show that, for $s>0$, $F$ is ...
0
votes
1answer
205 views
A mathematical notation for the mean value theorem?
I'm looking for a stringent definition of the mean value theorem i.e. stated with mathematical symbols. I think it should be something like "there exists..." and then the mean value if there is an ...
1
vote
0answers
42 views
What, in general, can I expect to be the restrictions and/or limitations to this alternative process to rewriting?
This is a specific idea I have to rewriting $x$ as $x y$, which I recently asked about in this question.
Suppose we have a power series
$$f(x) = \sum_{i=0}^\infty{c_i x^i} = c_0 x^0 + c_1 x^1 + c_2 ...
1
vote
2answers
68 views
Find the derivative of the primitive of a discontinous function
I have problems solving the following task:
I have the function $f(x)=\sin\frac{1}{x}$ when x isn't $0$ and $f(0) = 1$
First, I must prove that $f(x)$ is integrable in every interval $[a, b]$.
...
