Definite Integration
Definite Integration
nta_pyq_2025_jan
Grade 12
Question:
If \int 2 \pi 96x 2 cos 2 x dx = \pi (\alpha\pi 2 + \beta) , \alpha, \beta \in Z , then (\alpha + \beta) equals 2 - x (1+e ) 2
Step-by-Step Solution
Key Concept: Apply the core result for definite integral properties and simplify using the given constraints.
2 96x 2 cos 2 x I = \int dx (4) x 0 1 + e \pi 2 2 2 2I = 2 \int 96x cos xdx 0 \pi 2 2 2 I = 96 \int x cos xdx 0 \pi 2 2 = 48 \int x (1 + cos 2x)dx 0 \pi 2 2 = 2\pi + 48(0 - 0) - 48 \int x sin 2xdx 0 2 2 = 2\pi - 12\pi + [0 - 0] = \pi (2\pi - 12) 2 = \pi (\alpha\pi + \beta) 2 \Rightarrow (\alpha + \beta) = 100
Correct Answer: 4