EDB — 1BC

Exercises

1. [1BC]We define the Beta function as

   $$B(x,y)={\int }_0^1{t}^{x-1}(1-t{)}^{y-1}dt.$$

   1. Show that the integral exists (finite) if and only if $x,y>0$.

   2. Note that $B(x,y)=B(y,x)$

   3. Relate $B(n,m)$ to $B(n-1,m+1)$. Then calculate the value of $B(n,m)$ for $n,m$ natural positives.

   4. Use the result to calculate

      $${\int }_0^{𝜋/2}\sin (t{)}^9\cos (t{)}^{7}dt.$$
   Solution 1

   [1BD]↺↻