1. Ta có:
$I=\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sqrt{\cos x-\cos^3x}dx$
$=\int\limits_{-\frac{\pi}{2}}^{0}\sqrt{\cos x\sin^2x}dx+\int\limits_{0}^{\frac{\pi}{2}}\sqrt{\cos x\sin^2x}dx$
$=-\int\limits_{-\frac{\pi}{2}}^{0}\sqrt{\cos x}\sin xdx+\int\limits_{0}^{\frac{\pi}{2}}\sqrt{\cos x}\sin xdx$
Đặt $t=\cos x\Rightarrow dt=-\sin dx$
Ta có:
$I=\int\limits_0^1\sqrt tdt-\int\limits_1^0\sqrt tdt$
$=2\int\limits_0^1\sqrt tdt$
$=\dfrac{4}{3}\sqrt{t^3}\left|\begin{array}{l}1\\0\end{array}\right.=\dfrac{4}{3}$