jaunty stratus
Example:
$I_{\beta} \int_{0}^{1}{\frac{x-arcsinx}{x^{\beta}}}$,
$\beta \in \mathbb{R}$
Determine convergence of $I_{\beta}$. Calculate $I_2$ if it exists.
I don't know how to figure out singularities. In theory, I know they're points in the domain where the function reaches infinite value. Does that mean that I should take 2 limits (for endpoints of the interval) of the integrated function (and then determine if there's singularities depending on parameter $\beta$ probably)?
little nestBOT
danilojonic
subtle pecanBOT
Example:
$I_{\beta} \int_{0}^{1}{\frac{x-arcsinx}{x^{\beta}}}$,
$\beta \in...
+close
flat onyx
Example:
$I_{\beta} \int_{0}^{1}{\frac{x-arcsinx}{x^{\beta}}}$,
$\beta \in...
You forgot dx.
You need to see at which points the function under the integral becomes "problematic" (infinite limit or no limit). In this case that's only x = 0, so investigate what happens there.
dusky tinsel
**danilojonic**
there's just a point where integrand is not defined
$$ \lim _{t\to 0+} \int _t^1 f(x)dx $$
little nestBOT
aL
dusky tinsel
that's your task
flat onyx
there's just a point where integrand is not defined
I think that's too restrictive. Removeable and jump discontinuities shouldn't really matter.
little nestBOT
aL
dusky tinsel
dont even worry about integrating this thing analytically
if it converges, bound it from above by something convergent
otherwise from below by something divergent
jaunty stratus
for I2 specifically think about this
yeah i know this
dusky tinsel
or rather the integrand is negative, so absolute value
jaunty stratus
but examining the convergence is the only thing i didnt get really
flat onyx
Do we need to bound it? We can just analyze its behaviour.
jaunty stratus
okay so I should start with regular function examination?
dusky tinsel
if it's bounded in (0,1] then you're good
jaunty stratus
and then check for 0 and 1 critical points and asymptotes?
dusky tinsel
otherwise you need to get more specific
flat onyx
okay so I should start with regular function examination?
No, that's not what I meant.
Recall a couple of terms from the series of arcsin(x).
vestal cedar
to analyse the behaviour around 0 you can use Taylor expansion
jaunty stratus
to analyse the behaviour around 0 you can use Taylor expansion
yea thats what i thought too
dusky tinsel
it has a fairly simple antiderivative so that also works 
vestal cedar
$arcsin(x) \underset{x \to 0}{=} x+\frac{x^{3}}{6}+o(x^{3})$
little nestBOT
😑 rotoR
flat onyx
it has a fairly simple antiderivative so that also works <:noway:106089072265764...
It does?
Hm...
Are you sure?
I don't think arcsin(x)/x^a has an elementary antiderivative.
dusky tinsel
I2 case
flat onyx
Oh. Well, yeah, it does have one for certain values of a, but not for all.
dusky tinsel
regardless, taylor expansion is the usual goto method for such problems
$$ \frac{x-\arcsin x}{x^\alpha} =Cx^{3-\alpha} + O(x^4) $$
little nestBOT
aL
