If $f (x) = \frac{2 {sin}^{2} x + 2 sin x + 3}{{sin}^{2} x + sin x + 1}$ , then the number of integers in the range of $f (x)$ is

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Solution

$f (x) = \frac{2 {sin}^{2} x + 2 sin x + 3}{{sin}^{2} x + sin x + 1}$
$= 2 + \frac{1}{{sin}^{2} x + sin x + 1}$
$= 2 + \frac{1}{{(sin x + \frac{1}{2})}^{2} + \frac{3}{4}}$
$f (x)_{min} = 2 + \frac{1}{\frac{9}{4} + \frac{3}{4}} = \frac{7}{3}$
$f (x)_{max} = 2 + \frac{1}{3 / 4} = \frac{10}{3}$

Hence, range of $f (x)$ is $[\frac{7}{3}, \frac{10}{3}]$
$\Rightarrow$ Only one integral value i.e., $3$ .

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