Definite Integration
General
Grade 12
Question:
If $f(x) = \begin{cases} x+3 & : x < 3 \\ 3x^2+1 & : x \ge 3 \end{cases}$, then find $\int_2^5 f(x) dx$.
Step-by-Step Solution
Key Concept: General
$\int_2^5 f(x) dx = \int_2^3 f(x) dx + \int_3^5 f(x) dx = \int_2^3 (x+3) dx + \int_3^5 (3x^2+1) dx = \left[ \frac{x^2}{2} + 3x \right]_2^3 + \left[ x^3 + x \right]_3^5 = \frac{9-4}{2} + 3(3-2) + 5^3 - 3^3 + 5 - 3 = \frac{211}{2}$
Correct Answer: 211/2