Indefinite Integration
Indefinite Integration
nta_pyq_2025_jan
Grade 12
Question:
If f (x) = \int 1/4 1 1/4 dx, f (0) = -6 , then f (1) is equal to : x (1+x )
4 (log 2 - 2) e
2 - log e 2 2
log 2 + 2 e
4 (log 2 + 2) e
Step-by-Step Solution
Key Concept: Apply the core result for integration by substitution and identities and simplify using the given constraints.
Put x 1/4 = t \Rightarrow dx = 4t dt 3 3 2 (1) \int 4t dt = 4\int ( t - 1 + 1 ) dt t(t + 1) t + 1 t + 1 1/2 x 1/4 1/4 f (x) = 4 [ - x + ln∣ ∣x + 1∣ ∣] + C 2 f (0) = -6 \Rightarrow C = -6 \Rightarrow f (1) = 4 (log 2 - 2) e
Correct Answer: 1