Indefinite Integration 1 Question 20
21. The integral $\int _0^{\pi} \sqrt{1+4 \sin ^{2} \frac{x}{2}-4 \sin \frac{x}{2}} d x$ is equal to
(a) $\pi-4$
(b) $\frac{2 \pi}{3}-4-4 \sqrt{3}$
(c) $4 \sqrt{3}-4$
(d) $4 \sqrt{3}-4-\pi / 3$
(2014 Main)
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Solution:
- PLAN Use the formula, $|x-a|=\begin{array}{cc}x-a, & x \geq a \ -(x-a), & x<a\end{array}$
to break given integral in two parts and then integrate separately.
$$ \begin{aligned} & \int _0^{\pi} \sqrt{1-2 \sin \frac{x^{2}}{2}} d x=\int _0^{\pi}\left|1-2 \sin \frac{x}{2}\right| d x \\ & =\int _0^{\frac{\pi}{3}} 1-2 \sin \frac{x}{2} d x-\int _{\frac{\pi}{3}}^{\pi} 1-2 \sin \frac{x}{2} d x \\ & =x+4 \cos \frac{x}{2}{ } _0^{\frac{\pi}{3}}-x+4 \cos \frac{x}{2} \frac{\pi}{3} \\ & =4 \sqrt{3}-4-\frac{\pi}{3} \end{aligned} $$
