Discussion
The integrating factor of the differential equation \(sinx\ dy = \frac{1}{2}(sin2x + 2y\ cosx)dx\) is
Answer: E
The integrating factor is cosec x.
Since \(\frac{1}{2}\sin 2x = \sin x\cos x\), dividing by \(\sin x\,dx\) gives the linear form:
\[ \begin{aligned} \frac{dy}{dx} - (\cot x)\,y &= \cos x \\ \text{I.F.} &= e^{-\int\cot x\,dx} = e^{-\ln\sin x} = \frac{1}{\sin x} \end{aligned} \]
So the integrating factor is \(\csc x\). \(\sin x\) is what you get by forgetting the minus sign on the \(y\) term, and sec x, tan x and cos x come from integrating \(\cot x\) wrongly.
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