Let $f$ be a polynomial function such that $f\left(\mathrm{x}^{2}+1\right)=\mathrm{x}^{4}+5 \mathrm{x}^{2}+2$, for all $\mathrm{x} \in \mathbb{R}$. Then $\int_{0}^{3} f(\mathrm{x}) \mathrm{dx}$ is equal to

JEE Main · 2026 · 28 Jan 2026 · Shift 1 · Definite Integrals

Let

f

be a polynomial function such that

f (x^{2} + 1) = x^{4} + 5 x^{2} + 2

, for all

x \in R

.
Then

\int_{0}^{3} f (x) dx

is equal to

Answer: (B)

Putting

t = x^{2} + 1

(so

x^{2} = t - 1

f (t) = (t - 1)^{2} + 5 (t - 1) + 2 = t^{2} + 3 t - 2

. Then

\int_{0}^{3} f (x) d x = \int_{0}^{3} (x^{2} + 3 x - 2) d x = [\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - 2 x]_{0}^{3} = 9 + \frac{27}{2} - 6 = \frac{33}{2}

JEE Main · 2026 · 28 Jan 2026 · Shift 1 · Single correct option

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