Question

С подробным решением !

Answer 1

$boxed {1.} displaystyle int^{27}_0 frac{dx}{sqrt[3]{x^2}}=int^{27}_0 frac{1}{x^{2/3}}dx}=int^{27}_0 x^{-2/3}dx= frac{x^{-2/3+1}}{-frac{2}{3}+1}=$
$displaystyle = frac{x^{1/3}}{frac{1}{3}}=3 cdot x ^{1/3}= 3 sqrt[3] x bigg|^{27}_0= 3 sqrt[3] {27}-0=3 cdot 3 =9$

$boxed{2.} displaystyle int^{3 pi/2}_{pi/2} sin frac{x}{2}dx= frac{1}{ frac{1}{2}}int^{3 pi/2}_{pi/2} sin frac{x}{2}dx=2 cdot (-cos frac{x}{2})=-2cosfrac{x}{2} bigg|^{3 pi/2}_{pi/2}=$
$displaystyle = -2cos frac{3 pi}{2} cdot frac{1}{2}-(-2cos frac{ pi}{2} cdot frac{1}{2})=-2cos frac{3 pi}{4}-(-2cos frac{pi}{4})=$
$displaystyle = -(-2 cdot frac{ sqrt 2}{2})-(-2 cdot frac{ sqrt 2}{2})=sqrt2 + sqrt 2 = 2sqrt2$