Question

Помогите решить интегралы

Answer 1

$1) intlimits^4_1 frac{x^2-3x+1}{x^2} , dx = intlimits^4_1 (1-frac{3}{x}+x^{-2}), dx =\\=(x-3cdot ln|x|+frac{x^{-1}}{-1})Big |_1^4=(x-3cdot ln|x|-frac{1}{x})Big |_1^4=\\=(4-3ln4-frac{1}{4})-(1-3ln1-1)=3,75-3ln4-0=3,75-6ln2$

$2); ; intlimits^{frac{pi}{2}}_{frac{pi}{3}} (sinx+4cosx-1) , dx =(-cosx+4sinx-x)Big |_{frac{pi}{3}}^{frac{pi}{2}}=\\=(-cosfrac{pi}{2}+4sinfrac{pi}{2}-frac{pi}{2})-(-cosfrac{pi}{3}+4sinfrac{pi}{3}- frac{pi}{3} )=\\=0+4- frac{pi }{2} + frac{1}{2}-4cdot frac{sqrt3}{2}+ frac{pi }{3} =4,5-2sqrt3-frac{pi}{6}$

$3); ; intlimits^{frac{pi }{8}}_0 sin(2x+frac{pi}{4}) , dx =- frac{1}{2} cdot cos(2x+frac{pi }{4})Big |_0^{frac{pi}{8}} =\\=- frac{1}{2} cdot Big (cos frac{pi}{2} -cos frac{pi}{4}Big )=- frac{1}{2} cdot Big (0-frac{sqrt2}{2} Big ) =frac{sqrt2}{4}$