assume( alpha > 0, beta > 0, theta > 0, theta < 1 );

p1 := ( theta, alpha, beta ) -> theta^( alpha - 1 ) * 
  ( 1 - theta )^( beta - 1 );

                                        (alpha - 1)            (beta - 1)
     p1 := (theta, alpha, beta) -> theta            (1 - theta)

integrate( p1( theta, alpha, beta ), theta = 0 .. 1 );

                             Beta(alpha~, beta~)

help( Beta );

Beta - The Beta function
Calling Sequence:
     Beta( x, y )
Parameters:
     x - an expression 
     y - an expression 
Description:
- The Beta function is defined as follows: 
      Beta( x, y ) = ( GAMMA( x ) * GAMMA( y ) ) / GAMMA( x + y )

help( GAMMA );

GAMMA - The Gamma and Incomplete Gamma Functions
lnGAMMA - The log-Gamma function

Calling Sequence:

     GAMMA( z )
     GAMMA( a, z )
     lnGAMMA( z )
Parameters:
     z - an expression 
     a - an expression 
Description:

- The Gamma function is defined for Re( z ) > 0 by 
      GAMMA(z) =  int( exp( -t ) * t^( z - 1 ), t = 0 .. infinity )
  and is extended to the rest of the complex plane, 
  less the non-positive integers, by analytic continuation. 
  GAMMA has a simple pole at each of the points z = 0, -1, -2, ... .
- For positive real arguments z, the lnGAMMA function is defined by: 
      lnGAMMA( z ) = ln( GAMMA( z ) )

plotsetup( x11 );

plot( GAMMA( x ), x = 0 .. 5, color = black );