JEE Main 2025 Jan 24 Shift 1, Mathematics Q2: Some Properties of Definite Integrals

JEE Main2025Jan 24, Shift 1Mathematics

In $I (m, n) = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x$ , $m, n > 0$ , then $I (9, 14) + I (10, 13)$ is

Solution

The integral is the Beta function $B (m, n)$ , which satisfies the recurrence

$B (m, n) = B (m + 1, n) + B (m, n + 1)$ (split $1 = x + (1 - x)$ inside the integrand).

Apply this with $m = 9$ , $n = 13$ :

$I (9, 13) = I (10, 13) + I (9, 14)$ .

$∴$ Option (4).

Concept behind this question

Concept-wise practice with instant solutions on Fundamenthol.