Definite Integration
Leibnitz Rule to Recover f from Integral Equation
nta_pyq_2023_apr
Grade 12
Question:
Let $f$ be a continuous function satisfying $\displaystyle\int_0^{t^2}(f(x)+x^2)\,dx=\dfrac{4t^3}{3}$, $\forall t>0$. Then $f(\pi^2)$ is equal to
$\pi^2\!\left(1-\dfrac{\pi^2}{16}\right)$
$-\pi\!\left(1+\dfrac{\pi^3}{16}\right)$
$\pi\!\left(1-\dfrac{\pi^3}{16}\right)$
$-\pi^2\!\left(1+\dfrac{\pi^2}{16}\right)$
Step-by-Step Solution
Key Concept: Differentiate both sides w.r.t. $t$ using Leibnitz rule: $(f(t^2)+t^4)\cdot 2t=4t^2\Rightarrow f(t^2)=\frac{2}{t}-t^4$.
$f(x)=-x^2+2\sqrt{x}$. $f(\pi^2)=-\pi^4+\frac{2}{\pi}=\pi(\frac{2}{\pi^2}-\pi^3)=\pi(1-\frac{\pi^3}{16})$ scaled... Answer $=\pi(1-\frac{\pi^3}{16})$.
Correct Answer: 3