Question
Let f be a polynomial function such that f (x^ 2 +1 )=x^ 4 +5 x^ 2 +2, for all x R . Then _ 0 ^ 3 f(x) d x is equal to
JEE Main
Mathematics
Definite Integration
2026
JEE Main 2026 (28 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 5 3
- B. 41 3
- C. 27 2
- D. 33 2
Answer
D. 33 2
Step-by-step solution
Let u = x^2 + 1, then x^2 = u - 1. Substituting into the given condition: f(u) = (u-1)^2 + 5(u-1) + 2 = u^2 - 2u + 1 + 5u - 5 + 2 = u^2 + 3u - 2. Therefore, f(x) = x^2 + 3x - 2. Calculating the integral: _0^3 f(x)dx = _0^3 (x^2 + 3x - 2)dx = [ x^3 3 + 3x^2 2 - 2x ]_0^3 = 9 + 27 2 - 6 = 33 2 .
Related: Mathematics — Definite Integration · All PYQ Banks