Question

вычислить интегралы....

Answer 1

$1)\ intlimits^{2,25}_{0,25} {frac{dx}{sqrt{x}} =2sqrt{x}; |_{0,25}^{2,25}=2(sqrt{2,25}-sqrt{0,25})=2(1,5-0,5)=2$

$2)quad intlimits^2_0 {(4x^2-5x+1)} , dx =(4cdot frac{x^3}{3}-5cdot frac{x^2}{2}+x)|_0^2=\\= frac{4}{3}cdot 8- frac{5}{2} cdot 4+2= frac{8}{3} \\3)quad intlimits^{frac{pi}{2}}_{-frac{pi}{2}} {(cos^2x+1)} , dx = 2cdot intlimits^{frac{pi}{2}}_{0} {(frac{1+cos2x}{2}+1)} , dx=\\=2cdot intlimits^{frac{pi}{2}}_0 {(frac{3}{2}+frac{1}{2}cos2x)} , dx =(3x+frac{1}{2}sin2x)|_0^{frac{pi}{2}}=\\=frac{3pi}{2}+frac{1}{2}sinpi=frac{3pi}{2}$

$4)quad int limits _{frac{pi}{2}}^{frac{3pi}{2}}cos frac{x}{2} , dx=2cdot sinfrac{x}{2}; |_{frac{pi}{2}}^{frac{3pi}{2}}=2(sin frac{3pi}{4} -sinfrac{pi}{4} )=\\=2(frac{sqrt2}{2}-frac{sqrt2}{2})=0$