$S=\pi\displaystyle\int\limits^{\pi}_0 \sin^4x \, dx\\ =\pi I\\ I=\displaystyle\int\limits^{\pi}_0 (\sin^2x)^2 \, dx\\ =\displaystyle\int\limits^{\pi}_0 \left(\dfrac{1-\cos 2x}{2}\right)^2 \, dx\\ =\displaystyle\int\limits^{\pi}_0 \dfrac{(1-\cos 2x)^2}{4} \, dx\\ =\displaystyle\int\limits^{\pi}_0 \dfrac{1-2\cos 2x+\cos^22x}{4} \, dx\\ =\displaystyle\int\limits^{\pi}_0 \dfrac{1}{4} \, dx-\displaystyle\int\limits^{\pi}_0 \dfrac{\cos 2x}{2} \, dx+\displaystyle\int\limits^{\pi}_0 \dfrac{1+\cos 4x}{8} \, d(x)\\ =\displaystyle\int\limits^{\pi}_0 \dfrac{1}{4} \, dx-\displaystyle\int\limits^{\pi}_0 \dfrac{\cos 2x}{4} \, d(2x)+\displaystyle\int\limits^{\pi}_0 \dfrac{1+\cos 4x}{8} \, d(x)\\ =\dfrac{x}{4}\Bigg\vert^{\pi}_0-\dfrac{\sin 2x}{4} \Bigg\vert^{\pi}_0+\left(\dfrac{1}{8}x+\dfrac{\cos 4x}{32}\right)\Bigg\vert^{\pi}_0\\ =\dfrac{3\pi}{8}\\ \Rightarrow S=\dfrac{3\pi^2}{8}$
