Question

помогите решить примеры по интегралам срочно

Answer 1

$intlimits_1^2 {frac{2}{x^2}} , dx = 2 intlimits_1^2 {x^{-2}} , dx = -2x^{-1}|_1^2 = -frac{2}{x}|_1^2 = -frac{2}{2}+frac{2}{1x} = -1+2 = 1;$

$intlimits_1^2 {sqrt{x}} , dx = intlimits_1^2 {x^frac{1}{2}} , dx = frac{2}{3}x^frac{3}{2}|_1^2 = frac{2}{3}sqrt{x^3}|_1^2 = frac{2}{3}sqrt{2^3}-frac{2}{3}sqrt{1^3} = frac{4}{3}sqrt{2}-frac{2}{3};$

$intlimits_{-1}^0 {x-frac{1}{2}} , dx = intlimits_{-1}^0 {x} , dx - frac{1}{2}intlimits_{-1}^0 {} , dx = frac{x^2}{2}|_{-1}^0-frac{x}{2}|_{-1}^0 = frac{0^2}{2}-frac{(-1)^2}{2}-frac{0}{2}+frac{-1}{2}\=-frac{x}{2}-frac{x}{2}=-1;$

$intlimits_1^2 {x^{-3}} , dx = frac{x^{-2}}{-2}|_1^2 = -frac{1}{2x^2}|_1^2 = -frac{1}{2cdot2^2}+frac{1}{2cdot1^2} = -frac{1}{8}+frac{1}{2} = frac{3}{8};$

$intlimits_{-2}^{-1} {5x^3} , dx = frac{5x^4}{4}|_{-2}^{-1} = frac{5cdot(-2)^4}{4}-frac{5cdot(-1)^4}{4} = frac{5cdot(-2)^4}{4}-frac{5cdot(-1)^4}{4} = 20- frac{5}{4}=18,75$