Question

Let \(f(x)\) be a real valued function which is monotonic and differentiable. Then for any reals \(a\) and \(b\), \[ \int_{f(a)}^{f(b)}2x\{b-f^{-1}(x)\}\,dx= \]

• A. $\int_{a}^{b}(f^{2}(x)-f^{2}(a))dx$
• B. $\int_{a}^{b}(f(x)-f(a))^{2}dx$
• C. $\int_{a}^{b}(bf^{2}(x)-af^{2}(a))dx$
• D. $bf^{2}(b)+f^{-1}(a)$

Hint:

An elegant geometric way to view this problem is to visualize it as finding the area under a curve. Drawing out the bounding box regions created by the function transformations shows that the inverse integral blocks shift and balance each other out, collapsing the areas into this clean difference formula.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem uses the substitution rule for integrals involving inverse functions. We will substitute \( x = f(t) \) into the integral to express it in terms of the variable \( t \).
Step 2: Key Formula or Approach:
1. Substitution: \( x = f(t) \), then \( dx = f'(t) dt \).
2. Identity: \( f^{-1}(x) = f^{-1}(f(t)) = t \).
3. Integration by parts: \( \int u v' dt = uv - \int u' v dt \).
Step 3: Detailed Explanation:
Let \( I = \int_{f(a)}^{f(b)} 2x \{b - f^{-1}(x)\} dx \).
Put \( x = f(t) \). Limits: When \( x=f(a), t=a \); when \( x=f(b), t=b \).
\[ I = \int_{a}^{b} 2f(t) (b - t) f'(t) dt \]
Rearranging:
\[ I = \int_{a}^{b} (b - t) [2f(t) f'(t)] dt \]
Note that \( 2f(t) f'(t) = \frac{d}{dt}(f^2(t)) \).
Using integration by parts \( \int u dv \):
Let \( u = b - t \) and \( dv = \frac{d}{dt}(f^2(t)) dt \).
Then \( du = -dt \) and \( v = f^2(t) \).
\[ I = [(b - t)f^2(t)]_{a}^{b} - \int_{a}^{b} f^2(t) (-dt) \]
Evaluate the bracket at boundaries:
At \( t=b \): \( (b-b)f^2(b) = 0 \).
At \( t=a \): \( (b-a)f^2(a) \).
\[ I = [0 - (b - a)f^2(a)] + \int_{a}^{b} f^2(t) dt \]
\[ I = \int_{a}^{b} f^2(t) dt - \int_{a}^{b} f^2(a) dt \]
(Since \( (b-a)f^2(a) = \int_a^b f^2(a) dt \)).
\[ I = \int_{a}^{b} (f^2(t) - f^2(a)) dt \]
Replacing the dummy variable \( t \) with \( x \):
\[ I = \int_{a}^{b} (f^2(x) - f^2(a)) dx \]
Step 4: Final Answer:
The integral is equivalent to \( \int_{a}^{b} (f^2(x) - f^2(a))dx \).