Dat $x=sint , t\in [\frac{-\pi }{2};\frac{\pi }{2}]=>dx=costdt$
Doi can $x=0=>t=0$
$x=1=>\frac{\pi }{2}$
$=>I=\int\limits_{0}^{\frac{\pi }{2}}cos^{4}t.dt=\int\limits_{0}^{\frac{\pi}{2}}(\frac{cos2t+1}{2})^{2}dx=\int\limits_{0}^{\frac{\pi}{2}}\frac{cos4t}{8}dt+\int\limits_{0}^{\frac{\pi}{2}}\frac{cos2t}{2}dt+\int\limits_{0}^{\frac{\pi}{2}}\frac{5dt}{8}$
$I=\frac{5\pi}{16}$