$\int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} \mathrm{~d} x=$

$\int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} \mathrm{~d} x=$
  1. $2 \log \left(x+\sin \frac{x}{2}\right)+\mathrm{c}$, where c is a constant of integration.
  2. $\frac{1}{2} \log \left(x+\sin \frac{x}{2}\right)+\mathrm{c}$, where $\mathrm{c}$ is a constant of integration.
  3. $4 \log \left(x+\sin \frac{x}{2}\right)+c$, where c is a constant of integration.
  4. $\log \left(x+\sin \frac{x}{2}\right)+c$, where c is a constant of integration.

Solution

$\begin{aligned} & \text { Put } x+\sin \frac{x}{2}=\mathrm{t} \\ & \Rightarrow\left[1+\left(\cos \frac{x}{2}\right) \frac{1}{2}\right] \mathrm{d} x=\mathrm{dt} \\ & \Rightarrow\left(2+\cos \frac{x}{2}\right) \mathrm{d} x=2 \mathrm{dt}\end{aligned}$ $\therefore \quad \int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} \mathrm{~d} x=2 \int \frac{\mathrm{dt}}{\mathrm{t}}$ $\begin{aligned} & =2 \log |t|+c \\ & =2 \log \left|x+\sin \frac{x}{2}\right|+c\end{aligned}$

Asked in: MHT CET 2023 (13 May Shift 2)

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