First slide

Evaluation of definite integrals

Question

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

Moderate

A

$\frac{5}{4}$
B

7
C

4
D

2

Solution

$\begin{aligned} F (x) = \int_{0}^{x} f (t) d t \\ \Rightarrow F^{'} (x) = f (x) \\ ∴ f (4) = F^{'} (2) \end{aligned}$
$\begin{aligned} Also, 2 x F^{'} (x^{2}) = 2 x (1 + x) + x^{2} \\ \Rightarrow F^{'} (x^{2}) = 1 + 3 x / 2 \end{aligned}$
$∴ f (4) = F^{'} (2^{2}) = 1 + 3 = 4$

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