Question

решите пожалуйста. Все три

Answer 1

1-й пример:
$5x+1 equiv t, int {5sqrt{t}} , cdot dt = \ =5int {sqrt{t}} , cdot dt = 5 cdot {frac{2sqrt{t^3}}{3}} = {frac{10sqrt{t^3}}{3}} = {frac{10sqrt{(5x+1)^3}}{3}} = \ intlimits^3_0 {sqrt{5x+1}} , dx = {frac{10sqrt{(5x+1)^3}}{3}} |^{3}_{0}= \ ={frac{10sqrt{(5 cdot 3+1)^3}}{3}} - {frac{10sqrt{(5 cdot 0 +1)^3}}{3}} = frac {640}{3}-frac{10}{3}=\ =frac{630}{3}=210$
2-й пример:
$int {cos 2x} , dx = frac {sin 2x}{2} \ intlimits^{frac{ pi }{3}}_{-frac{pi}{6}} {cos 2x} , dx = frac {sin 2x}{2} |^{frac{ pi }{3}}_{-frac{pi}{6}} = \ = frac {sin frac{ 2 cdot pi }{3}}{2} - frac {sin frac{ 2 cdot pi }{6}}{2} = \ = frac {-1}{2} - frac {frac {sqrt{3}}{2}}{2} = -frac{2+sqrt{3}}{4}$
3-й пример:
$int {x cdot e^x} , dx = e^x cdot (x-1) +C \ intlimits^1_0 {x cdot e^x} , dx = e^x cdot (x-1)|^{1}_{0} = e^1 cdot (1-1) - e^0 cdot (0-1) = \ = e cdot 0 - 1 cdot (-1) = 0 + 1 = 1$