Inte-greater!

Relevant wiki: Integration Tricks

Write numerator as $\small{\frac 12\cdot 2\sin x=\frac 12\left((\sin x-\cos x)+(\sin x+\cos x)\right)}$ so that integration $(\mathfrak I)$ is:

$\mathfrak I=\dfrac 12\displaystyle\int _{ - \pi / 6 }^{ \pi / 6 }\left(1+\dfrac{\overbrace{\sin x+\cos x}^{\mathrm{d}(\sin x-\cos x)}}{\sin x-\cos x}\right)\mathrm{d}x$

$=\dfrac{\pi}{6}+\left[\dfrac{1}2\ln|\sin x-\cos x|\right] _{ - \pi / 6 }^{ \pi / 6 }$

$=\frac \pi 6 + \frac 12 \ln \left(2-\sqrt 3\right)$

Hence, $\Large 6+2+3=\boxed{11}$

$\begin{aligned} I & = \int_{-\frac \pi 6}^\frac \pi 6 \frac {\sin x}{\sin x - \cos x} dx & \small \color{#3D99F6}{\text{By }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx } \\ & = \int_{-\frac \pi 6}^\frac \pi 6 \frac {\sin (-x)}{\sin (-x) - \cos (-x)} dx \\ & = \int_{-\frac \pi 6}^\frac \pi 6 \frac {\sin x}{\sin x + \cos x} dx \\ \implies 2I & = \int_{-\frac \pi 6}^\frac \pi 6 \frac {\sin x}{\sin x - \cos x} dx + \int_{-\frac \pi 6}^\frac \pi 6 \frac {\sin x}{\sin x + \cos x} dx \\ & = \int_{-\frac \pi 6}^\frac \pi 6 \frac {\sin x(\sin x + \cos x + \sin x - \cos x)}{\sin^2 x - \cos^2 x} dx \\ & = \int_{-\frac \pi 6}^\frac \pi 6 \frac {2\sin^2 x}{\sin^2 x - \cos^2 x} dx \\ & = \int_{-\frac \pi 6}^\frac \pi 6 \frac {1-\cos 2x}{-\cos 2x} dx \\ & = \int_{-\frac \pi 6}^\frac \pi 6 (1-\sec 2x) \ dx \\ & = x \bigg|_{-\frac \pi 6}^\frac \pi 6 - 2\int_0^\frac \pi 6 \sec 2x \ dx \\ & = \frac \pi 3 - 2\times \frac 12 \ln (\tan 2x + \sec 2x) \ \bigg|_0^\frac \pi 6 \\ & = \frac \pi 3 - \ln (\sqrt 3 + 2) + \ln (0+1) \\ & = \frac \pi 3 + \ln \left(\frac 1{\sqrt 3 + 2}\right) \\ & = \frac \pi 3 + \ln \left(2-\sqrt 3\right) \\ \implies I & = \frac \pi 6 + \frac 12 \ln \left(2-\sqrt 3\right)\end{aligned}$

$\implies A+B+C = 6+2+3 = \boxed{11}$

I don't know how to do this in LaTeX but $A + B + C = 6 + 2 + 3 = 11$

Numerator can be rewritten as 0.5(sinx-cosx)+0.5(sinx+cosx) split the denominator and use small substitution to find answer orally

split into interval -pi/6 to 0 and 0 to pi/6 . Then after basic manipulation we get 2sin^2 x in num and -cos2x in denominator .Add and subtract cos^2 x in numerator