JEE Main 2025 (Online) 24th January Morning ShiftQuestion 1
If $I(m,n)=\int_0^1x^{m-1}(1-x)^{n-1}dx$, $m,n>0$, then $I(9,14)+I(10,13)$ is
Definite Integration
Solution
Concept. Both integrands share $x^8(1-x)^{12}$; the leftover factors add up to $1$. Formulas used. (1) $I(m,n)=\int_0^1x^{m-1}(1-x)^{n-1}dx$Step 1 - Write both. $$I(9,14)=\int_0^1x^8(1-x)^{13}dx,\qquad I(10,13)=\int_0^1x^9(1-x)^{12}dx.$$ Step 2 - Add. $$I(9,14)+I(10,13)=\int_0^1x^8(1-x)^{12}\big[(1-x)+x\big]dx=\int_0^1x^8(1-x)^{12}dx=I(9,13).$$ Answer: (D)