Integrating factor of \(\displaystyle\frac{dy}{dx}+y\cot x=2x+x^2\cot x\ (x \ne 0)\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\sin x\)
The equation is linear, \(\frac{dy}{dx} + Py = Q\), with \(P = \cot x\). Its integrating factor is \(e^{\int P\,dx}\):
\( \displaystyle \int\cot x\,dx = \log|\sin x| \)
\(\text{I.F.} = e^{\log|\sin x|} = \sin x\)
(taking \(\sin x \gt 0\), as is usual for the integrating factor).