Definite Integration
Definite Integration
nta_pyq_2025_jan
Grade 12
Question:
Let f : R \to R be a twice differentiable function such that f (2) = 1. If F(x) = xf (x) for all x \in R, and \int , then F (2) + \int is equal to : 2 ′ 2 ′′ ′ 2 2 \int x F (x)dx = 6 x F (x)dx = 40 F(x)dx 0 0 0
Step-by-Step Solution
Key Concept: Apply the core result for definite integral properties and simplify using the given constraints.
2 ′ \int xF (x)dx = 6 (1) 0 2 2 = xF(x)| - \int f (x)dx = 6 0 0 2 = 2 F(2) - \int xF(x)dx = 6[\therefore f (2) = 2 F(2) = 2] 0 2 \int xF(x)dx = -2. . . (1) 0 2 \Rightarrow \int F(x)dx = -2. . . (2) 0 Also 2 2 2 2 ′′ 2 ′ ′ ′ \int x F (x)dx = x F (x)∣ ∣ - 2\int x F (x)dx = 40 0 0 0 ′ = 4F (2) - 2 \times 6 = 40 ′ F (2) = 13 2 ′ \therefore F (2) + \int F (x) = 13 - 2 = 11 0 \pi
Correct Answer: 1