Let f be a polynomial function such that f(x² + 1) = x⁴ + 5x² + 2
Q. Let f be a polynomial function such that f(x2 + 1) = x4 + 5x2 + 2, for all x ∈ ℝ.

Then ∫03 f(x) dx is equal to

(A) 33/2

(B) 5/3

(C) 27/2

(D) 41/3

Correct Answer: 33/2

Explanation

Given,

f(x2 + 1) = x4 + 5x2 + 2

Let,

x2 + 1 = t ⇒ x2 = t − 1

Then,

f(t) = (t − 1)2 + 5(t − 1) + 2

f(t) = t2 − 2t + 1 + 5t − 5 + 2

f(t) = t2 + 3t − 2

Hence,

f(x) = x2 + 3x − 2

Now evaluate the integral:

∫03 (x2 + 3x − 2) dx

= [ x3/3 + 3x2/2 − 2x ]03

= (27/3 + 27/2 − 6)

= 9 + 13.5 − 6

= 33/2

Hence, the required value is 33/2.

Related JEE Main Questions

Let A, B and C be three 2 × 2 matrices with real entries such that B = (I + A)⁻¹ and A + C = I. If BC = [1 −5; −1 2] and CB [x₁; x₂] = [12; −6], then x₁ + x₂ is A bag contains 10 balls out of which k are red and (10 − k) are black. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x + 2√2y = 4. If the coordinates of the vertex A are (α, β), then the greatest integer less than or equal to |α + √2β| is The value of lim x→0 loge(sec(ex)·sec(e²x)·…·sec(e¹⁰x)) divided by e² − e²cosx is equal to Let y = y(x) be the solution of the differential equation x dy/dx − sin2y = x³(2 − x³)cos²y, x ≠ 0. If y(2) = 0, then tan(y(1)) is equal to

Updated for JEE Main 2026: This PYQ is important for JEE Mains, JEE Advanced and other competitive exams. Practice more questions from this chapter.

Scroll to Top