# Thread: what gets done to f(t) in this integral?

1. ## what gets done to f(t) in this integral?

$\displaystyle \arcsin{(t)}=\int\frac{1}{\sqrt{1-t^2}}dt$

Does this mean that...

$\displaystyle \arcsin{(f(t))}=\int\frac{1}{\sqrt{1-f(t)^2}}dt$ ?

Or what happens to f(t) when this integral is evaluated? Does it just stay the same? That would be nice if it did, but I imagine something complicated happens to it.

2. Suppose to start from...

$\displaystyle \int \frac{dt}{\sqrt{1-t^{2}}}= \sin^{-1} t + c$ (1)

... and that we want to insert under the integral sign $\displaystyle x=f(t)$ instead of $\displaystyle t$. It will be...

$\displaystyle \frac{dx}{dt} = f^{'} (t) \rightarrow dx= f^{'}(t)\cdot dt$ (2)

... so that we obtain...

$\displaystyle \int \frac{dx}{\sqrt {1-x^{2}}}= \int \frac{f^{'}(t)}{\sqrt{1-f^{2}(t)}}\cdot dt = \sin^{-1} \{f(t)\} + c$ (3)

Merry Christmas from Italy

$\displaystyle \chi$ $\displaystyle \sigma$

3. Think of it in terms of something simpler. You know that $\displaystyle \int x dx= \frac{1}{2}x^2+ C$. But $\displaystyle \int f(x) dx$ is NOT equal to $\displaystyle \frac{1}{2} f^2(x)+ C$ for every function f! That would make integration trivial. What you are doing is essentially a substitution. And if you let u= f(x), then you also have to let du= f'(x)dx.

4. Originally Posted by chisigma
Suppose to start from...

$\displaystyle \int \frac{dt}{\sqrt{1-t^{2}}}= \sin^{-1} t + c$ (1)

... and that we want to insert under the integral sign $\displaystyle x=f(t)$ instead of $\displaystyle t$. It will be...

$\displaystyle \frac{dx}{dt} = f^{'} (t) \rightarrow dx= f^{'}(t)\cdot dt$ (2)

... so that we obtain...

$\displaystyle \int \frac{dx}{\sqrt {1-x^{2}}}= \int \frac{f^{'}(t)}{\sqrt{1-f^{2}(t)}}\cdot dt = \sin^{-1} \{f(t)\} + c$ (3)

$\displaystyle \chi$ $\displaystyle \sigma$

Does the $\displaystyle f^{2}(t)$ in equation 3 mean "f of t squared" or "second derivative of f(t)"?

5. Is...

$\displaystyle f(t)\cdot f(t) = f^{2} (t)$ (1)

... and...

$\displaystyle \frac{d^{2}f}{dt^{2}} = f^{(2)} (t)$ (2)

Merry Christmas from Italy

$\displaystyle \chi$ $\displaystyle \sigma$