0
votes
0answers
11 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
1
vote
1answer
23 views

Application of the mean value theorem for Integrals

Suppose that $f(x)$ is a differentiable function in $[a,b]$, $f^{'}(x)$ is a monotone decreasing function in $(a,b)$, and $f^{'}(b)>0$. So how to prove that $$ \big \vert \int_a^b \cos ...
1
vote
2answers
112 views

Differentiating with respect to the limit of integration

I'm confused about problems involving differentiation with respect to the limit of an integral, I just want to check that my understanding is correct. For example, are the following statements ...
0
votes
0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
2
votes
1answer
31 views

Differentiability of the convolution $\int_0^tf(t-s)g(s)\;ds$

Given two continuously differentiable functions $f,g:[0,\infty)\to\mathbb{R}$. I want to know what we can tell about the differentiability of $$(f\ast g)(t)=\int_0^tf(t-s)g(s)\;ds$$ Especially, why ...
1
vote
1answer
38 views

Smoothing Lemma

Given a $C^0$ function $g:[a,b]\to \mathbb{R}$ that is smooth everywhere except at $c$ (where $a<c<b$), and has positive derivative everywhere except at $c$, the claim is that there exists a ...
0
votes
0answers
14 views

Integration by parts with Legendre Functions

I need help deriving $\int_{-l}^l [P_l^m(x)]^2 = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$ for the associated Legendre functions I am supposed to use $P_l^m(x) = (-1)^{-m}\int_{-l}^l ...
5
votes
3answers
132 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
4
votes
1answer
48 views

Given the following derivatives, find the integrals

Find the derivatives of $\ln(x+\sqrt{x^2+1})$ and $\arcsin(x)$, and use the result to find the integrals of the following functions: $$ \dfrac{1}{ \sqrt{ \pm x^2 \pm a^2 }} $$ $$ ...
0
votes
1answer
27 views

Derivatives and Integrals of Summations

Im unsure if this is just a stupid question because i have been independently studying this kind of math for about a week, but this has been bothering me lately as i have been exploring some definite ...
1
vote
1answer
22 views

Differential equation of inclined plane

I'm having some trouble with the equation $$\frac{d}{dt}\dot{x}=g\sin\Theta \implies \dot{x}(t)=\dot{x}(t=0)+\int_0^t dt'\:g\sin\Theta=\dot{x_0}+g\:t\sin\Theta $$ which appears in page 4 of ...
0
votes
0answers
29 views

n integrals in summation

I've seen it might be possible to write a summation that looks like $$ \sum\limits_{i=1}^{\infty}\left\{\frac{\partial}{\partial x_i}\left(\frac{xy}{\sqrt{x^2+y^2}}\right)\right\} $$ But what about ...
0
votes
0answers
13 views

Integration : Green's Function in estimating displacement of non-prismatic beams

I'm working on a non-prismatic structure similar to that in Figure 3 of Page 10 (345) from an article entitled: "Green’s function for the deflection of non-prismatic simply supported beams by an ...
0
votes
1answer
28 views

Differentiating a function and using the result to calculate the indefinite integral of another.

We should differentiate the function $f(x) = \sqrt{cosx}$ and use the result to calculate the indefinite integral $\int \frac{sinx}{\sqrt{cosx}}dx$. So I started by differentiating $f(x) = ...
1
vote
0answers
32 views

Integration by Parts in matrices

Given Data in the question We have a given equation based on matrices as follows $\frac{\mathrm{d} R(s)_{3\times3}}{\mathrm{d} s}=R(s)_{3\times3}K(s)_{3\times3} \tag 1$ $\frac{\mathrm{d} ...
7
votes
1answer
63 views

Evaluating $\int x^n e^{x}dx$

I consider, for $n=0,1,2,...$, $$ u_n(x)=\int x^n e^{x}dx.$$ I've performed an integration by parts giving $$ u_n(x)=nx^{n-1} e^{x}-nu_{n-1}(x).$$ I'm looking for a closed form. Thank you.
0
votes
1answer
22 views

Change of Variable involve derivative

Let me just give the 1-D version of my problem. Let $u\in C_c^\infty(R)$ and define $u_r(x):=u(rx)$. Then I am trying to evaluate the integration $\int_R u_r'(x)dx$. Here is my steps: $$\int_R ...
0
votes
1answer
50 views

Integral using Beta Function and Gamma Function

Interestingly, I seem to have an integral I have posted before, but I want to take a different approach to it. $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} \,dx$ The beta function states, $B(x,y) = ...
3
votes
1answer
195 views

Calculus integral evaluation using substitution

I have to find this integral: Evaluate the integral using an appropriate substitution $$\int\dfrac{8e^x+7e^{-x}}{8e^x-7e^{-x}}\mathrm dx.$$ I've tried my solution $\ln\Big[15\cdot \sinh(x) + ...
1
vote
0answers
41 views

derivative of normalizer in exponential form — change integral and gradient

When deriving the relation between normalizer and expectation of the sufficient statistic for distributions in exponential form one uses the fact, that the density integrates to one: $$1 = ...
2
votes
1answer
25 views

Total Time Taken Question

Distance of chord = Time taken to "swim" to the desalination plant = I'm stuck here! The textbook working out is as such: I don't understand how they have the 'k' or 1/2 the runs river at ...
0
votes
1answer
23 views

Shortest Chord from origin to function

worked solution: Is this found using the distance of a line equation, where instead of co-ordinate points they use functions, so the two functions are g(x) and x (because the origin is on the ...
0
votes
2answers
63 views

Integrate $\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$ [closed]

Evaluate$$\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$$ Solution through either contour integral method or indefinite integral method please!
1
vote
2answers
47 views

Intermediate step for integration?

Among the various method of Integration there is one specific method(it may vary according to the terms) where for instance if we have a function as:$$\frac{px+q}{ax^2+bx+c}$$ To integrate this we ...
2
votes
0answers
34 views

Differentiation - a technical point

I understand the following equation to be correct, but why can we treat the differentials as fractions and cancel them out? What would be the correct way to view it? $$ \int_{-\pi/a} ...
1
vote
0answers
17 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
1
vote
2answers
82 views

What do we mean by derivative of a function? What does it tell? [duplicate]

Taking the derivative of any kind of function is easy but I don't know why we take the derivative? Like $f(x)=x^2$ has the derivative $2x$, so what does it mean? I don't know how to define ...
2
votes
0answers
15 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
1
vote
0answers
36 views

Proving that maximal interval of existence exists and that solution is unque

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
0
votes
1answer
68 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
1
vote
2answers
47 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
1
vote
1answer
79 views

Does a word problem provide all information?

A while ago I asked a similar question about word problems and assumptions. Is it a definition or an accepted-fact that word problems provide all information about the relevant existence/situation in ...
0
votes
0answers
13 views

Differential and a notation problem

Let $df(x) = f'(x)\,dx \:\:\:\:\: (1)$ Now we want to integrate both sides and we get: $f(x) = f(x)$ But now we want to differentiate again and we get $f'(x)=f'(x)$ I just don get it. If we say ...
3
votes
1answer
64 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
0
votes
0answers
27 views

Integration with matrices

I have written two equations in matrix format as follows $m(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} ...
2
votes
2answers
78 views

Is there any geometric explanation of relationship between Integral and derivative?

It is said integral is anti-derivative, derivative is tangent of graph function in each point on the function and integral is the area of the region in the xy-plane bounded by the graph. I can not ...
4
votes
6answers
82 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
1
vote
1answer
44 views

First and second derivatives of the function $f(x)=x\int_0^x e^{t^2}dt$

I haven't done calculus for a while so I need your help with these two exercises. I am not sure whether my solutions are correct so I'd really appreciate someone's feedback. $$ f(x)=x\int_0^x ...
1
vote
2answers
65 views

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then ...
0
votes
0answers
20 views

Error of Riemann sum is $a/n + o(1/n)$ [duplicate]

A problem from an old qual: For $f$ of class $C^2$, find $a$ such that $$\int_0^1 f(t)dt-\frac1n\sum_{k=1}^{n-1}f\left( \frac {k}{n}\right)=\frac{a}{n}+o\left(\frac1n \right).$$ If we divide ...
1
vote
2answers
58 views

How to prove that a derivative of a formula equals to another formula.

If $u= \ln(\tan x+\tan y+\tan z)$ prove $$\sin 2x \dfrac{du}{dx} + \sin 2y \dfrac{du}{dy} + \sin 2z \dfrac{du}{dz}=2 $$ My answwer was like this: $$u' =\dfrac{ 1}{\tan x+\tan y+\tan z} \cdot( ...
1
vote
2answers
43 views

Integrable combinations - I can't seem to arrive at the given answer

I need help! I can't seem to arrive at the answer given in our textbook. I'm new here, so I really need help. The instruction says that I need to solve this D.E by recognizing integrable ...
3
votes
1answer
41 views

Calc II - Definite integral of sqrt(t^2 + t) from 2x to 1?

How do I find $$\int_1^{2x}\sqrt{t^2 + t}$$ with only knowledge from a Calculus I course? I've tried plugging this puppy into Wolfram Alpha and other integral solvers, which report it as solvable ...
2
votes
3answers
157 views

Limit of a Riemann Sum and Integral

I've been trying to solve this problem, but I haven't been able to calculate the exact limit, I've just been able to find some boundaries. I hope you guys can help me with it. Let $f:[0,1] \to ...
1
vote
0answers
92 views

Problem with trigonometric substitution proof

I'm sad, I can't get it. I know perfectly how to integrate using the mechanical process described in the books, but I want to understand the proof of it. My book (Stewart) says: In general we can ...
1
vote
1answer
66 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
0
votes
0answers
26 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
1
vote
1answer
36 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
0
votes
1answer
31 views

Derivative of an integral with variable in upper bound and a term of the integrand

So I want to take the first and second derivatives of a function g(Z) which is made up of several terms, one of which is where Z and H are our variables. Taking the derivative of this, it seems ...
3
votes
3answers
75 views

Find the limit and derivative of integral function.

$\psi_m(x)$ is defined as $$\int_0^{\ln|x|}e^{mt}\sin(t)^m\mathop{dt}$$ with $m$ a natural number greater then zero. Now the question is, does $\lim\limits_{x\to 0}\psi_m(x)$ exist. I've tried using ...