Definite Integration
General
Grade 12
Question:
If $f(x)$ is integrable function such that $|f(x) - f(y)| \le |x^2 - y^2|, \forall \ x, y \in [a, b]$ then prove that $\left| \int_a^b \frac{f(x) - f(a)}{x + a} dx \right| \le \frac{(a - b)^2}{2}$.
Step-by-Step Solution
Key Concept: General
Given, $\left| \int_a^b \frac{f(x) - f(a)}{x + a} dx \right| \le \int_a^b \left| \frac{f(x) - f(a)}{x + a} \right| dx \le \int_a^b \left| \frac{x^2 - a^2}{x + a} \right| dx = \int_a^b |x - a| dx = \int_a^b (x - a) dx = \frac{(a - b)^2}{2}$
Correct Answer: A