First slide

Evaluation of definite integrals

Question

If $f (x) = \int_{1 / x^{2}}^{x^{2}} \cos \sqrt{t} d t,$ then $f^{'} (1)$ is equal to

Moderate

A

$\cos 1$
B

$2 \cos 1$
C

$4 \cos 1$
D

none of these

Solution

$f^{'} (x) = \cos \sqrt{x^{2}} \frac{d}{d x} (x^{2}) - \cos \sqrt{1 / x^{2}} \frac{d}{d x} \frac{1}{x^{2}}$
(Using Property 18)
$= 2 x \cos x + \frac{2}{x^{3}} \cos (1 / x) .$ Hence $f^{'} (1) = 4 \cos 1 .$

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