If $y = y (x)$ is the solution curve of the differential equation
$\frac{dy}{dx} + ytan x = xsec x, 0 \leq x \leq \frac{π}{3}, y (0) = 1, then y (\frac{π}{6})$ is equal to

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$\frac{π}{12} - \frac{\sqrt{3}}{2} \log_{e} (\frac{2 \sqrt{3}}{e})$

$\frac{π}{12} + \frac{\sqrt{3}}{2} \log_{e} (\frac{2}{e \sqrt{3}})$

$\frac{π}{12} - \frac{\sqrt{3}}{2} \log_{e} (\frac{2}{e \sqrt{3}})$

$\frac{π}{12} + \frac{\sqrt{3}}{2} \log_{e} (\frac{2 \sqrt{3}}{e})$

answer is A.

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Detailed Solution

Integrating factor = sec x
$\begin{array}{l} y (\sec x) = xtan x - l n (\sec x) + c \\ y (o) = 1 \Rightarrow c = 1 \\ y (\sec x) = xtan x - l n (\sec x) + 1 \\ x = \frac{π}{6}, y = \frac{π}{12} + \frac{\sqrt{3}}{2} l n (\frac{\sqrt{3}}{2}) + \frac{\sqrt{3}}{2} l n_{e}^{e} \\ y = \frac{π}{12} + \frac{\sqrt{3}}{2} l n (\frac{\sqrt{3} e}{2}) \\ y = \frac{π}{12} - \frac{\sqrt{3}}{2} l n (\frac{2}{\sqrt{3} e}) \end{array}$