Question

Let \( f \) be a polynomial function such that \[ f(x^2+1)=x^4+5x^2+2,\quad \text{for all } x\in\mathbb{R}. \] Then \[ \int_0^3 f(x)\,dx \] is equal to:

• A. \( \dfrac{5}{3} \)
• B. \( \dfrac{27}{2} \)
• C. \( \dfrac{33}{2} \)
• D. \( \dfrac{41}{3} \)

Hint:

When a polynomial identity involves an expression like \( f(x^2+1) \), substitute \( t=x^2+1 \) to convert it into a standard polynomial comparison problem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to determine the functional form of \(f(x)\) from the given relation and then integrate it.
Step 2: Detailed Explanation:
Given \(f(x^2 + 1) = x^4 + 5x^2 + 2\).
Let \(t = x^2 + 1\). Then \(x^2 = t - 1\).
Substitute \(x^2\) into the expression for \(f\):
\[ f(t) = (t - 1)^2 + 5(t - 1) + 2 \] \[ f(t) = (t^2 - 2t + 1) + 5t - 5 + 2 = t^2 + 3t - 2 \] So, \(f(x) = x^2 + 3x - 2\).
Now, evaluate the integral:
\[ \int_{0}^{3} (x^2 + 3x - 2) dx = \left[ \frac{x^3}{3} + \frac{3x^2}{2} - 2x \right]_{0}^{3} \] \[ = \left( \frac{27}{3} + \frac{3(9)}{2} - 2(3) \right) - 0 \] \[ = 9 + \frac{27}{2} - 6 = 3 + 13.5 = 16.5 = \frac{33}{2} \] Step 3: Final Answer:
The value of the integral is 33/2.