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

Question

Let $f$ be a continuous function satisfying $\int _{0}^{{t}^{2}}(f(x)+{x}^{2})dx=\frac{4}{3}{t}^{3},\forallt>0$. Then

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

Accepted Answer

Given that

$\int _{0}^{{t}^{2}}(f(x)+{x}^{2})dx=\frac{4}{3}{t}^{3},\forallt>0$

On differentiating using Leibnitz rule, we get

$\begin{gathered} (f({t}^{2})+{t}^{4})	imes 2t=\frac{4}{3}	imes 3{t}^{2} \ \Rightarrow f({t}^{2})+{t}^{4}=2t \ \Rightarrow f({t}^{2})=2t-{t}^{4} \end{gathered}$

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

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

Solutions · 1