4
votes
6answers
80 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
42 views

Finding the area bounded by two curves when in terms of $x = y^2$?

I can't seem to figure this problem out. Find the area bounded by the curves $x=2y-y^2$ and $x=4-y^2$, in the first quadrant. I am having difficulties with graphing the equations and coming up ...
0
votes
1answer
116 views

Normalization of a two-dimensional kernel function

I've got three two-dimensional kernel functions which look like this $$ k(r,h) = n \cdot \begin{cases} \ldots & 0 \le r \le h \\ 0 & otherwise \end{cases} $$ With ...
1
vote
1answer
38 views

How to integrate the inverse of sin

How does one integrate $\int\ {\sin^{-1}(x)}$, using integration by parts, where: $$ u={\sin^{-1}}, du=\frac{1}{\sqrt{1-x^2}},dv=dx, v=x ? $$ This is a partial solution, and I do not quite ...
0
votes
2answers
43 views

Inverse functions determination by integral

From "Inverse functions and differentiation": Integrating this relationship gives $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. $$ This is only useful if the integral exists. ...
1
vote
2answers
31 views

Could someone please help me find the derivative of the inverse of $f$ at $0$?

The problem is: for $\displaystyle f(x)= \int_0^{\ln x} \frac{1}{\sqrt{4+\mathrm{e}^{t}}} \, \mathrm{d}t$, $x > 0$, find $(f^{-1})'(0)$. I know that I should use the fundamental theorem of ...
2
votes
1answer
22 views

Inverse integral manipulation

Why is the following true: If $\rho = B^{-1}(t)$ where $$ B(t) = \int_0^t \frac{1}{g(\gamma(s))}ds $$ then $$ t = \int_0^\rho \frac{1}{g(\gamma(s))}ds $$ I know it must be something fundamental, ...
1
vote
1answer
67 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
2
votes
0answers
43 views

Integrating inverse functions

I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...
1
vote
1answer
49 views

How to calculate the inverse of the line integeral.

Let $f$ be a polynomial function, $$ f(x) = a_0 + a_1 x + ... + a_d x^d $$ where $a_0$, $a_1$, ..., $a_d$ are parameters and usually $d \le 6$. Let $g$ be the line integral of $f$, $$ g(x) = ...
1
vote
0answers
55 views

Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...
2
votes
0answers
53 views

About the functional inverse of integrals and infinite products.

It seems $\cos(x)$ and $\sin(x)$ are the only entire functions, that are the functional inverse of an integral of some elementary function $f(x)$ , such that they have a simple infinite product ...
1
vote
0answers
41 views

Getting an inverse function

I have a cubic function $N_3(x) = a x^3 + b x^2 + c x + d$ which guaranteed is non-negative in each point on interval $x \in [0,1]$. I building an other function $N_4(x) = \int_0^x{N_3(t)dt}$. Sure ...
4
votes
2answers
119 views

$\ln(x)$, $e^{x}$ and $\int \frac{1}{x}dx$ relationship

My math professor told me that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$ by the definition; so far so good. But how/why does $\ln(x)$ ($\int_1^x\frac{1}{t} dt$: by defintion) coincide with the inverse of ...
4
votes
0answers
182 views

Inversion of elliptic integral

I have an equation of the type $$ p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx, $$ in which $a$ and $b$ (with $a>b>0$) are (known) functions of some parameter $H$ (such that it is ...
0
votes
0answers
178 views

Can one find the inverse function for a combination of imcomplete gamma functions?

The original function was defined as $f(z)=y=\left( \Gamma \left( k,{\frac {z-b}{\theta}} \right) -\Gamma \left( k,{\frac {z-a}{\theta}} \right) \right) \left( \Gamma \left( k \right) \right) ...
5
votes
1answer
408 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...