Computing $\int_{0 }^{1} (1-x^{\frac{1 } {a}})^{-\frac{1 } {2 }} dx$ I was wondering how to compute directly this integral without using beta/gamma functions:
$\int_{0 }^{1} (1-x^{\frac{1 } {a}})^{-\frac{1 } {2 }} dx$, 
$a\to 0$
Wolfram Alpha said its equal to 1, but I can't get it by direct calculation.
I tried the substitution :
$t=x^{\frac{1} {a}} $ so $t^a=x$ then $dx=at^{a-1}dt$ 
But failed to calculate  the new integral. 
Thank you 

I think the limit does not exist. only for $a\rightarrow 0^+$ the limit exists. 
In first step With out  using beta function I show
$$lim_{a\rightarrow 0^+} \int_{0 }^{1} (1-x^{\frac{1 } {a}})^{-\frac{1 } {2 }} dx=1$$ 
$a$ must be positive since
$(1-x^{\frac{1 } {a}})^{-\frac{1 } {2 }}=\frac{1}{(1-x^{\frac{1}{a}})^{\frac{1}{2}}}$
so $1-x^{\frac{1}{a}}>0$ so $a>0$. so this limit  is valid only for $a\rightarrow 0^+$
x<-.5
a<--.0001
(1-x^(1/a))^(-.5) #NaN

if $0<x<1$  so $0<x^{\frac{1}{a}}<1$ for $a> 0$
$$lim_{a\rightarrow 0^{+}} \int_{0 }^{1} (1-x^{\frac{1 } {a}})^{-\frac{1 } {2 }} dx$$
$$=lim_{n\rightarrow +\infty} \int_{0 }^{1} (1-x^n)^{-\frac{1 } {2 }} dx$$
define $0\leq f_n=(1-x^n)^{-\frac{1 } {2 }}=\frac{1}{(1-x^n)^{\frac{1 } {2 }}}$
 so $f_n$ are decreasing and non-negative so by monotone-convergence-theorem $lim \int f_n=\int lim f_n$
$$\overset{MCT}{=}
 \int_{0 }^{1} lim_{n\rightarrow +\infty} (1-x^n)^{-\frac{1 } {2 }} dx$$
since $0<x<1$ so $x^{n} \rightarrow 0$
$$=\int_{0 }^{1} 1 dx=1$$
. 
With beta function
 let $a> 0$
$t=x^{\frac{1} {a}} $ so $t^a=x$ then $dx=at^{a-1}dt$ 
$$=\int_{0 }^{1} (1-t)^{-\frac{1 } {2 }} at^{a-1}dt$$
$$=a\int_{0 }^{1} (1-t)^{\frac{1 } {2 }-1} t^{a-1}dt$$
 so $a> 0$ Beta_distribution (this is why $a\rightarrow 0^+$)
$$=a B(a,\frac{1}{2})=a\frac{\Gamma(a)\Gamma(\frac{1}{2})}{\Gamma(a+\frac{1}{2})}$$
$$=\frac{\Gamma(a+1)\Gamma(\frac{1}{2})}{\Gamma(a+\frac{1}{2})}
\rightarrow 1 \hspace{1cm} when \, \, a\rightarrow 0^+ $$
Rcode 
by a -->0+
a<<-.00001
fun<-function(x){
(1-x^(1/a))^(-.5)
}
integrate(fun,lower=0,upper=1)  ### 1 with absolute error < 1.1e-14
by a -->0-
a<<--.00001
integrate(fun,lower=0,upper=1)  
#Error in integrate(fun, lower = 0, upper = 1) : non-finite function value

