If /(m, n)=∫ x^(m-1) (1-x)^(n-1) dx, x ∈ to (0, 1) m, n>0, then /(9, 14) + /(10, 13) is equal to

Question

If \(I(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n>0,\) then \(/(9,14)+/(10, 13)\) is equal to

(1) \(/(1,13)\)

(2) \(/(9,1)\)

(3) \(/(9,13)\)

(4) \(/(19,29)\)
 

Answer 1

Correct option is (3) \(/(9,13)\)     

Beta function  

\(\beta(p, q)=\int_0^1 x^{p-1}(1-x)^{q-1} d x=\frac{(p-1)!(q-1)!}{(p+q-1)!} p, q \in I\)   

\(\Rightarrow /(9,14)=\frac{8!13!}{22!} \)

\(/(10,13)=\frac{9!\cdot12!}{22!} \)

\(\Rightarrow /(9,14)+/(10,13)=\frac{1}{22!}(8!13!+9!\cdot 12!) \)

\(\Rightarrow=\frac{1}{22!} 8!\cdot12!(13+9)=\frac{8!12!}{21!} \)

\(= \frac{(9-1)!(13-1)!}{(9+13-1)!}=/(9,13)\)