Question

Решить определённые интегралы (пошагово)

Answer 1

1. $intlimits^1_0 {frac{x}{x^2-5x+6} } , dx = frac{1}{2} intlimits^1_0 {frac{1}{x^2-5x+6} } , dx^2 =  frac{1}{2} intlimits^1_0 {frac{1}{x^2-5x+frac{25}{4}+6-frac{25}{4}  } } , dx^2 = frac{1}{2} intlimits^1_0 {frac{1}{(x-frac{5}{2})^2-frac{1}{4}  } } , dx^2 = frac{1}{2} ln|(x-frac{5}{2})^2-frac{1}{4}  | |frac{1}{0} = frac{1}{2} (lnfrac{9}{4} -frac{1}{4} -lnfrac{25}{4} +frac{1}{4} ) = frac{1}{2} (lnfrac{9}{4} - lnfrac{25}{4} ) = frac{1}{2} lnfrac{9*4}{4*25} =$ $frac{1}{2} lnfrac{9}{25}$

2. $intlimits^e_1 {x^2log(x)} , dx = |u=log(x), dv = x^2dx, du = frac{1}{xlna} dx, v = frac{x^3}{3} | = frac{log(x)x^3}{3} - intlimits^e_1 {frac{x^3}{3xlna} } , dx = frac{log(x)x^3}{3} - frac{1}{3lna} intlimits^e_1 {x^2 } , dx = frac{log(x)x^3}{3} - frac{1}{3lna} *frac{x^3}{3} |frac{e}{1} = frac{log(e)e^3}{3} - frac{1}{3lna} * frac{e^3}{3} + frac{1}{3lna} *frac{1}{3} = frac{log(e)e^3}{3} - frac{1}{3lna} (frac{e^3-1}{3} )$