Indefinite Integration
Indefinite Integration
nta_pyq_2025_jan
Grade 12
Question:
Let \int x sin x dx = g(x) + C , where C is the constant of integration. If 3 8 (g ( \pi 2 ) + g ( ′ \pi 2 )) = \alpha\pi 3 + \beta\pi 2 + \gamma, \alpha, \beta, \gamma \in Z , then \alpha + \beta - \gamma equals :
Step-by-Step Solution
Key Concept: Apply the core result for integration by substitution and identities and simplify using the given constraints.
\int x 3 sin xdx = -x 3 cos x + \int 3x 2 cos xdx (2) 3 2 = -x cos x + 3x sin x - \int 6x sin xdx 3 2 = -x cos x + 3x sin x + 6x cos x - 6 sin x + c So g(x) = -x cos x + 3x sin x + 6x cos x - 6 sin x 3 2 2 \pi 3\pi g( ) = - 6 2 4 ′ 2 3 g (x) = -3x cos x + x sin x + 6 cos x - 6 cos x 3 \pi \pi ′ g ( ) = 2 8 \pi \pi ′ 3 2 8( g( ) + g ( )) = \pi + 6\pi - 48 2 2 So \alpha + \beta - \gamma = 55
Correct Answer: 2