Calculate $f (x) + f^{'} (x) + 2$ , if $f (x) = x sin 2 x$

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Solution

Here, Given $f (x) = x sin 2 x$
$f^{'} (x) = \frac{d}{d x} (x sin 2 x)$
$= x \frac{d}{d x} (sin 2 x) + sin 2 x \frac{d}{d x} (x)$ [Product Rule of differentiation]
$= 2 x cos 2 x + sin 2 x (1)$
$= 2 x cos 2 x + sin 2 x$
So, $f (x) + f^{'} (x) + 2 = x sin 2 x + 2 x cos 2 x + sin 2 x + 2$
$= sin 2 x (x + 1) + 2 x cos 2 x + 2$
$= 2 x cos 2 x + sin 2 x (x + 1) + 2$ .

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