Let f be a continuous function satisfying [_0^t^2(f(x)+x^2) d x=(4)/(3) t^3, t 0].

Question

Let $f$ be a continuous function satisfying $$\int_\limits{0}^{t^{2}}\left(f(x)+x^{2}\right) d x=\frac{4}{3} t^{3}, \forall t > 0$$. Then

$$f\left(\frac{\pi^{2}}{4}\right)$$ is equal to :

Accepted Answer

Given that

$$
\int\limits_0^{t^2}\left(f(x)+x^2\right) d x=\frac{4}{3} t^3, \forall t>0
$$

On differentiating using Leibnitz rule, we get

$$
\begin{aligned}
& \left(f\left(t^2\right)+t^4\right) \times 2 t=\frac{4}{3} \times 3 t^2 \\
& \Rightarrow f\left(t^2\right)+t^4=2 t \\
& \Rightarrow f\left(t^2\right)=2 t-t^4
\end{aligned}
$$

On substituting $\frac{\pi}{2}$ for $t$, we get

$$
f\left(\frac{\pi^2}{4}\right)=2\left(\frac{\pi}{2}\right)-\frac{\pi^4}{16}=\pi\left(1-\frac{\pi^3}{16}\right)
$$

Solutions · 1