Question

If \(f(x^2 + 1) = x^4 + 5x^2 + 1\), then find \(\int_{0}^{3} f(x) dx\) :

• A. 13.5
• B. 15.3
• C. 13
• D. 15.5

Hint:

When dealing with \( f(g(x)) \), identifying \( f(x) \) first makes integration much more straightforward than doing substitution inside the integral.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to find the explicit expression for \( f(x) \) using substitution.
Step 2: Key Formula or Approach:
Let \( x^2 + 1 = t \implies x^2 = t - 1 \).
Step 3: Detailed Explanation:
Substitute \( x^2 = t - 1 \) into the equation for \( f(x^2+1) \):
\( f(t) = (t-1)^2 + 5(t-1) + 1 \)
\( f(t) = (t^2 - 2t + 1) + 5t - 5 + 1 \)
\( f(t) = t^2 + 3t - 3 \).
So, \( f(x) = x^2 + 3x - 3 \).
Evaluate the integral:
\[ I = \int_{0}^{3} (x^2 + 3x - 3) dx = \left[ \frac{x^3}{3} + \frac{3x^2}{2} - 3x \right]_0^3 \] \[ I = \left( \frac{27}{3} + \frac{27}{2} - 9 \right) - (0) = 9 + 13.5 - 9 = 13.5 \] Step 4: Final Answer:
The value of the integral is 13.5.