Definite Integration
Integral Calculus-2
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f(x)$ be a continuous function such that $f(x) > 0$ for all $x \geq 0$ and $\left(f(x)\right)^{101} = 1 + \int_0^x f(t) dt$, then $(f(10))^{100}$ is equal to ____.
Step-by-Step Solution
Key Concept: Differentiate the functional equation to reduce the power and create an integrable form.
Given $(f(x))^{101} = 1 + \int_0^x f(t) dt$, differentiate both sides to get $101(f(x))^{100} f'(x) = f(x)$. This simplifies to $101(f(x))^{99} f'(x) = 1$ (assuming $f(x) > 0$). Integrating: $\frac{101(f(x))^{100}}{100} = x + C$. Using $f(0) = 1$, we find $C = \frac{101}{100}$. Therefore $\frac{101(f(x))^{100}}{100} = x + \frac{101}{100}$. At $x = 101$: $(f(101))^{100} = 101$, so $f(101) = 101^{1/100}$.
Correct Answer: 101