Indefinite Integration
Indefinite Integration
nta_pyq_2025_jan
Grade 12
Question:
Let I(x) = \int 11 dx 15 . If I(37) - I(24) = 1 4 ( 1 1 - 1 1 ) , b, c \in N , then 3( b + c) is equal to (x-11) 13 (x+15) 13 b 13 c 13
Step-by-Step Solution
Key Concept: Apply the core result for integration by substitution and identities and simplify using the given constraints.
I (x) = \int dx 11 15 (2) (x - 11) 13 (x + 15) 13 x - 11 26 Put = t \Rightarrow dx = dt 2 x + 15 (x + 5) 2/13 1 dt 1 t I (x) = \int = ⋅ 26 t 11/13 26 2/13 2/13 1 x - 11 I(x) = ( ) + C 4 x + 15 2/13 2/13 1 26 1 13 I(37) - I(24) = ( ) - ( ) 4 52 4 39 1 1 1 = ( - ) 2/13 4 2/13 2 3 1 1 1 = ( - ) 4 1/13 1/13 4 9 \therefore b = 4, c = 9 3( b + c) = 39
Correct Answer: 2