First slide

Evaluation of definite integrals

Question

Let $f$ be a continuous function satisfying $\int_{- π}^{t^{2}} (f (x) + x^{2}) d x = π + \frac{4}{3} t^{3}$ for all t, then $f (π^{2} / 4)$ is equal to

Moderate

A

$π - \frac{π^{4}}{8}$
B

$\frac{π}{2} - {(\frac{π}{4})}^{4}$
C

$\frac{π}{2} - {(\frac{π}{4})}^{2}$
D

$π - \frac{π^{4}}{16}$

Solution

Differentiating both sides, we have
$\begin{array}{l} (f (t^{2}) + t^{4}) 2 t = 4 t^{2} \\ f (t^{2}) + t^{4} = 2 t \\ \Rightarrow f (t^{2}) = 2 t - t^{4} \end{array}$
so $f (\frac{π^{2}}{4}) = f ({(\frac{π}{2})}^{2}) = 2 \frac{π}{2} - {(\frac{π}{2})}^{4}$
$= π - \frac{π^{4}}{16}$ .

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