Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f: [0, \infty) \to \mathbb{R}$ be a continuous strictly increasing function such that $f'(x) = \int_{0}^{x} tf^2(t)dt$ for every $x \geq 0$, then value of $f(6)$ is _____.
Step-by-Step Solution
Key Concept: The quotient of two proportional functions is constant, making its derivative zero.
Given $u(x) = 7v(x) ⟹ u'(x) = 7v'(x)$, we have $p = 7$. Dividing both sides by $v(x)$ gives $\frac{u(x)}{v(x)} = 7 ⟹ \left[\frac{u(x)}{v(x)}\right]' = 0$, so $q = 0$. Therefore $\frac{p+q}{p-q} = \frac{7+0}{7-0} = 1$.
Correct Answer: 6