Question

ПОМОГИТЕ ПОЖАЛУЙСТА 6 ИНТЕГРАЛОВ 100 БАЛЛОВ!!!!!!!!!!!!!!!

Answer 1

(1)
$intlimits {9x^2+5cos(x)} , dx = 9*intlimits {x^2} , dx +5* intlimits {cos(x)} , dx =\\ =9*frac{x^3}{3}+5*sin(x)+C=3x^3+5sin(x)+C$

(2)
$intlimits {frac{5x^4-6}{x}} , dx = intlimits {5x^3-frac{6}{x}} , dx =\\ =5* intlimits {x^3} , dx -6* intlimits^a_b {frac{1}{x}} , dx =\\ =5*frac{x^4}{4}-6*ln|x|+C=\\ =frac{5}{4}x^4-6*ln|x|+C$

(3)
$intlimits {frac{x}{sqrt{3-2x^2}}} , dx = intlimits {frac{2x}{2sqrt{3-2x^2}}} , dx = intlimits {frac{1}{2sqrt{3-2x^2}}} , d(x^2) =\\= intlimits {frac{1}{-4sqrt{3-2x^2}}} , d(-2x^2)=intlimits {frac{1}{-4sqrt{3-2x^2}}} , d(3-2x^2)=\\ =[t=3-2x^2]=-frac{1}{4}* intlimits {frac{1}{sqrt{t}}} , dt=-frac{1}{4}* intlimits {t^{-frac{1}{2}}} , dt=\\ =-frac{1}{4}*frac{t^{frac{1}{2}}}{frac{1}{2}}+C=-frac{sqrt{t}}{2}+C=-frac{sqrt{3-2x^2}}{2}+C$

(4)
$intlimits {frac{(xsqrt{x}-3)^2}{x^3}} , dx = intlimits {frac{x^3-6x^{frac{3}{2}}-9}{x^3}} , dx = intlimits {1} , dx -6 intlimits^a_b {x^{-frac{3}{2}}} , dx - 9*intlimits {x^{-3}} , dx =\\ =x-6*frac{x^{-frac{1}{2}}}{-frac{1}{2}}-9*frac{x^{-2}}{-2}+C=\\ =x+frac{12}{sqrt{x}}+frac{9}{2x^2}+C$

(5)
$intlimits {x^2(1+2x)} , dx = intlimits {x^2+2x^3} , dx = intlimits {x^2} , dx +2* intlimits {x^3} , dx =\\=frac{x^3}{3}+2*frac{x^4}{4}+C= frac{x^3}{3}+frac{x^4}{2}+C$

(6)
$intlimits {e^{-3x}} , dx = intlimits {frac{e^{-3x}}{-3}} , d(-3x) =[t=-3x]=\\=-frac{1}{3}* intlimits {e^t} , dt=-frac{1}{3}e^t+C=\\ =-frac{1}{3}e^{-3x}+C$