Q113 / 180 · KCET 2025 · Mathematics

The value of $\int_0^{2 \pi} \sqrt{1+\sin \left(\frac{x}{2}\right)} d x$ is

  • Correct answer
    A
    $8$
  • B
    $4$
  • C
    $2$
  • D
    $0$

Solution

$\begin{aligned} & \int_0^{2 \pi} \sqrt{1+\sin \frac{x}{2}} \\ & \begin{aligned} \int_0^{2 \pi}\left|\cos \frac{x}{4}+\sin \frac{x}{4}\right| & =\left[4 \sin \frac{x}{4}-4 \cos \frac{x}{4}\right]_0^{2 \pi} \\ & =(4(1)-0(0-4) \\ & =8\end{aligned}\end{aligned}$

Hence, the correct answer is option: 'A'.

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