Let f be a continuous function satisfying _0^ t^2 ( f ( x )+ x ^2 ) dx =4 3 t ^3, t > 0 . Then f ( ^2 4 ) equal to :

Let f be a continuous function satisfying _0^ t^2 ( f ( x )+ x ^2…

MCQ

Let $f$ be a continuous function satisfying $\int \limits_0^{t^2}\left( f ( x )+ x ^2\right) dx =\frac{4}{3} t ^3, \forall t > 0 . \quad$ Then $f \left(\frac{\pi^2}{4}\right)$ equal to :

✓
$\pi\left(1-\frac{\pi^3}{16}\right)$
B
$-\pi^2\left(1+\frac{\pi^2}{16}\right)$
C
$-\pi\left(1+\frac{\pi^3}{16}\right)$
D
$\pi^2\left(1-\frac{\pi^2}{16}\right)$

✓

Answer

Correct option: A.

$\pi\left(1-\frac{\pi^3}{16}\right)$

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

$\left( f \left( t ^2\right)+ t ^4\right)=2 t$

$f \left( t ^2\right)=2 t - t ^4$

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

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

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