Let f be a polynomial function such that f ( x 2 + 1 ) = x 4 + 5 x 2 + 2 , for all x . Then 0 3 f ( x ) ? d x is equal to: [2026]

Question

Let f be a polynomial function such that $f (x^{2} + 1) = x^{4} + 5 x^{2} + 2, for all x \in ℝ .$ Then $\int_{0}^{3} f (x) d x$ is equal to: [2026]

Smart Achievers · Accepted Answer

(2)

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

$Put x^{2} + 1 = t$

$\Rightarrow f (t) = {(t - 1)}^{2} + 5 (t - 1) + 2$

$\Rightarrow f (t) = t^{2} + 3 t - 2$

$Now, \int_{0}^{3} f (t) d t = \int_{0}^{3} (t^{2} + 3 t - 2) d t$

$= {[\frac{t^{3}}{3} + \frac{3 t^{2}}{2} - 2 t]}_{0}^{3}$

$= [\frac{27}{3} + \frac{27}{2} - 6]$

$= \frac{33}{2}$

Solution

Q.
Let f be a polynomial function such that $f (x^{2} + 1) = x^{4} + 5 x^{2} + 2, for all x \in ℝ .$ Then $\int_{0}^{3} f (x) d x$ is equal to: [2026]

1 $\frac{27}{2}$

2 $\frac{33}{2}$

3 $\frac{41}{3}$

4 $\frac{5}{3}$

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let f be a polynomial function such that f(x2+1)=x4+5x2+2, for all x∈ℝ. Then ∫03f(x) dx is equal to: [2026]

1 272

2 332

3 413

4 53