Definite Integration
Definite Integration
nta_pyq_2025_jan
Grade 12
Question:
If I(m, n) = \int 1 0 x m-1 (1 - x) n-1 dx, m, n > 0 , then I (9, 14) + I (10, 13) is
I(19, 27)
I(9, 1)
I (1, 13)
I(9, 13)
Step-by-Step Solution
Key Concept: Apply the core result for definite integral properties and simplify using the given constraints.
1 m-1 n-1 I(m, m) = \int x (1 - x) dx (4) 0 2 Let x = sin \theta dx = 2 sin \theta cos \theta d\theta \pi/2 2 m-1 2n-1 I(m, n) = 2 \int (sin \theta) (cos \theta) d\theta 0 \pi/2 17 27 I(9, 14) + I(10, 13) = 2 \int (sin \theta) (cos \theta) d\theta 0 \pi/2 19 25 + 2\int (sin \theta) (cos \theta) d\theta 0 \pi/2 17 25 = 2\int (sin \theta) (cos \theta) d\theta 0 = I(9, 13)
Correct Answer: 4