Definite Integration
Definite Integration
nta_pyq_2025_jan
Grade 12
Question:
Let for some function y = f (x), \int and f (2) = 3. Then f (6) is equal to x 2 tf (t)dt = x f (x), x > 0 0
Step-by-Step Solution
Key Concept: Apply the core result for definite integral properties and simplify using the given constraints.
\int tf (t)dt = x f (x) 2 (1) 0 2 ′ \Rightarrow xf (x) = 2xf (x) + x f (x) 2 ′ \Rightarrow x ⋅ f (x) = -xf (x) ′ f (x) 1 \Rightarrow = - f (x) x ′ f (x) 1 \int = -\int dx f (x)dx x \Rightarrow ln f (x) = - ln x + C f (2) = 3 \Rightarrow ln 3 = - ln 2 + C C = ln 6 6 \Rightarrow f (x) = \Rightarrow f (6) = 1 x
Correct Answer: 1