Question

2π\3
∫ (sin x\4+cos x\4)^2 dx
0

Answer 1

$\displaystyle\\\int_{0}^{\frac{2\pi}{3}}\left(\sin\dfrac{x}{4}+\cos\dfrac{x}{4}\right)^2\mathrm{d}x=\int_{0}^{\frac{2\pi}{3}}\left(1+\sin\dfrac{x}{2}\right)\mathrm{d}x=\int_{0}^{\frac{2\pi}{3}}1\,\mathrm{d}x+\int_{0}^{\frac{2\pi}{3}}\sin\dfrac{x}{2}\,\mathrm{d}x=\bigskip\\=x\Big|_{0}^{\frac{2\pi}{3}}+\left(-2\cos\dfrac{x}{2}\right)\Big|_{0}^{\frac{2\pi}{3}}=\left(\dfrac{2\pi}{3}-0\right)+\left(-2\cos\dfrac{2\pi}{6}-\left(-2\cos 0\right)\right)=$

$=\dfrac{2\pi}{3}+\left(-2\cdot \dfrac{1}{2}+2\right)=\dfrac{2\pi}{3}+(2-1)=\dfrac{2\pi}{3}+1$