What is the integrating factor of the differential equation $\frac{dy}{dx}+y\tan x=\sec x$?

Solution

1. The given differential equation is in the standard linear form $\frac{dy}{dx}+P(x)y=Q(x)$, where $P(x)=\tan x$.
2. The integrating factor is defined by the formula $$\begin{array}{rl}I(x)& =e^{\int P(x)dx}\\ & =e^{\int \tan xdx}\end{array}$$.
3. Evaluating the integral gives $e^{\ln |\sec x|}$, which simplifies to $\sec x$.
4. Hence the answer is (A).