Discussion
The integrating factor of the differential equation \((x+1)\dfrac{dy}{dx} - y = \sin x\) is
Answer: D
The integrating factor is \(1/(x+1)\).
Divide by \(x + 1\) to put the equation in the standard linear form \(dy/dx + P(x)\,y = Q(x)\):
\[ \begin{aligned} \frac{dy}{dx} - \frac{y}{x+1} &= \frac{\sin x}{x+1} \\ P(x) &= -\frac{1}{x+1} \\ \text{IF} &= e^{\int P\,dx} = e^{-\ln(x+1)} \\ &= \frac{1}{x+1} \end{aligned} \]
Taking \(x + 1\) comes from forgetting the minus sign in \(P(x)\). The options \(x\) and \(1/x\) would belong to an equation with \(x\), not \(x + 1\), as the coefficient of \(dy/dx\).
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