The integral _0^ [1+4^2x2]^1/2dx equals: *(interpreting garbled OCR as _0^ √1+4^2(x/2),dx …

Question

The integral $\displaystyle\int_0^\pi \left[1+4\sin^2\dfrac{x}{2}\right]^{1/2}dx$ equals: *(interpreting garbled OCR as $\int_0^\pi \sqrt{1+4\sin^2(x/2)}\,dx$ appears inconsistent with the worked solution below, which evaluates $\int_0^\pi \left|2\sin\frac{x}{2}-1\right|dx$)*

Accepted Answer

Let $I = \int_0^\pi \left|2\sin\dfrac{x}{2}-1\right|dx = \int_0^{\pi/3}\left(1-2\sin\dfrac{x}{2}\right)dx + \int_{\pi/3}^\pi \left(2\sin\dfrac{x}{2}-1\right)dx$

(since $\sin\dfrac{x}{2}=\dfrac12$ at $x=\dfrac{\pi}{3}$)

$= \left[x+4\cos\dfrac{x}{2}\right]_0^{\pi/3} + \left[-4\cos\dfrac{x}{2}-x\right]_{\pi/3}^{\pi}$

$= \dfrac{\pi}{3}+4\cdot\dfrac{\sqrt3}{2}-4+0-\pi+4\cdot\dfrac{\sqrt3}{2}+\dfrac{\pi}{3} = 4\sqrt3-4-\dfrac{\pi}{3}$

Solutions · 1