Definite Integration
Definite Integration
nta_pyq_2025_apr
Grade 12
Question:
Let for some function $y = f(x)$, $\displaystyle\int_0^x tf(t)\,dt = x^2 f(x)$, $x>0$ and $f(2) = 3$. Then $f(6)$ is equal to
Step-by-Step Solution
Key Concept: Differentiate both sides of $\int_0^x tf(t)dt = x^2 f(x)$ with respect to $x$ to obtain $xf(x) = 2xf(x)+x^2f'(x)$, then separate variables to solve the resulting ODE $f'/f = -1/x$.
Differentiating both sides:
$$xf(x) = 2xf(x) + x^2 f'(x) \Rightarrow x^2 f'(x) = -xf(x).$$
$$\frac{f'(x)}{f(x)} = -\frac{1}{x} \Rightarrow \ln f(x) = -\ln x + C \Rightarrow f(x) = \frac{K}{x}.$$
$f(2) = 3 \Rightarrow K = 6$. Hence $f(x) = \dfrac{6}{x}$ and $f(6) = 1$.
Correct Answer: 1