# Tagged Questions

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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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, ...
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### 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 $t^n$)?
I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps ...