KCET · Maths · Definite Integration
The value of \(\int_0^{2 \pi} \sqrt{1+\sin \left(\frac{x}{2}\right)} d x\) is
- A \(8\)
- B \(4\)
- C \(2\)
- D \(0\)
Answer & Solution
Correct Answer
(A) \(8\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int_0^{2 \pi} \sqrt{1+\sin \frac{x}{2}} \\ & \begin{aligned} \int_0^{2 \pi}\left|\cos \frac{x}{4}+\sin \frac{x}{4}\right| & =\left[4 \sin \frac{x}{4}-4 \cos \frac{x}{4}\right]_0^{2 \pi} \\ & =(4(1)-0(0-4) \\ & =8\end{aligned}\end{aligned}\)
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