Solve pi int _ { pi / 3 } ^ frac { 2 pi { 3 } } ( sin x ) ^ 2 wrt x + 2 pi ∫ (from 0 to frac { pi) of { 3 } } ( sin 2 x ) ^ 2 dx

$$\pi \int _ { \frac { \pi } { 3 } } ^ { \frac { 2 \pi } { 3 } } ( \sin x ) ^ { 2 } d x + 2 \pi \int _ { 0 } ^ { \frac { \pi } { 3 } } ( \sin 2 x ) ^ { 2 } d x$$

Factor

$$\frac{\pi {(3 \sqrt{3} + 4 \pi)}}{8} = 6.975326485308174$$

Solution Steps

Factor out \pi .

$$\pi \left(\int _{\frac{\pi }{3}}^{\frac{2\pi }{3}}\left(\sin(x)\right)^{2}\mathrm{d}x+2\int _{0}^{\frac{\pi }{3}}\left(\sin(2x)\right)^{2}\mathrm{d}x\right)$$

Evaluate

$$\frac{\pi \left(3\sqrt{3}+4\pi \right)}{8}\approx 6.975326485$$