First slide

Evaluation of definite integrals

Question

Let $f : (0, \infty) \to R$ and $F (x) = \int_{0}^{x} f (t) d t .$ If $F (x^{2}) = x^{2} (1 + x)$ then $f (4)$ equals

Moderate

A

5/4
B

7
C

4
D

2

Solution

$F (x^{2}) = \int_{0}^{x^{2}} f (t)$ dt , there fore, $x^{2} (1 + x) =$
$\int_{0}^{x^{2}} f (t) d t$ . Differentiating both sides w.r.t. x using Property
17, we have
$2 x + 3 x^{2} = f (x^{2}) \cdot 2 x \Rightarrow f (x^{2}) = 1 + (3 / 2) x$
Putting $x = 2$ we have $f (4) = 1 + 3 = 4$ .

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