The integral equals \int_{0}^{\pi} \sqrt{1 + 4 \left(sin\right)^{2} \frac{x}{2}

Engineering

Mathematics

Properties of Definite Integral

Question

The integral equals $\int_{0}^{π} \sqrt{1 + 4 \sin^{2} \frac{x}{2} - 4 \sin \frac{x}{2}} dx$

$\frac{2 π}{3} - 4 - 4 \sqrt{3}$

$4 \sqrt{3} - 4 - \frac{π}{3}$

$4 \sqrt{3} - 4$

π - 4

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$\int_{0}^{π} | 1 - 2 \sin \frac{x}{2} | dx$

Put, $\frac{x}{2}$ = t ⇒ dx = 2dt

$2 \int_{0}^{\frac{π}{2}} | 1 - 2 sint | dt$

$= 2 (\int_{0}^{\frac{π}{6}} (1 - 2 sint) dt + \int_{\frac{π}{6}}^{\frac{π}{2}} (2 sint - 1) dt)$

$= 4 \sqrt{3} - 4 - \frac{π}{3}$