Question

Помогите, пожалуйста, решить примеры, не понимаю...

Answer 1

$int {frac{dx}{sqrt{8-2x}} } = | 8-2x = t^2 => 2tdt = -2dx => dx = -tdt| = int {frac{-tdt}{sqrt{t^2}} } = -int {dt} } = t + c = sqrt{8-2x} + c\int {frac{6x-5}{3x^2-5x+4} } }dx = | 3x^2-5x+4 = t => dt = (6x-5)dx| = int {frac{dt}{t} } } = ln|t| + c = ln(3x^2-5x+4) + lnC = lnC(3x^2-5x+4)$

$int {frac{arcsin^3(2x)}{sqrt{1-4x^2}} }dx = |arcsin(2x) = t => dt = frac{2dx}{sqrt{1-4x^2}} => frac{dt}{2} = frac{dx}{sqrt{1-4x^2}}| = int {frac{t^3*frac{dt}{2} }{1} } = int{frac{1}{2}t^3dt } = frac{1}{8}t^4 + c =frac{1}{8}arcsin^4(2x) + c$

$int{x*arccos(2x)dx} = |arccos(2x) = u => du = -frac{2dx}{sqrt{1-4x^2}}. dv = xdx => v = frac{1}{2}x^2| = frac{1}{2}x^2*arccos(2x) + int{frac{2*frac{1}{2}x^2}{sqrt{1-4x^2}}dx = frac{1}{2}x^2*arccos(2x) + int{frac{x^2}{sqrt{1-4x^2}}dx$

$int{frac{x^2}{sqrt{1-4x^2}}dx = | x = frac{1}{2}sin(t) => dx = frac{1}{2}cos(t)dt| = int{frac{0.25sin^2t * 0.5cos(t)}{cos(t)}dt = frac{1}{8}int{sin^2tdt} =$$frac{1}{16} int{(1-cos2t})dt = frac{1}{16}(t - frac{1}{2}sin2t)+c = |x = 0.5sin(t) => t = arcsin(2x)| = frac{1}{16}(arcsin(2x) - frac{1}{2}sin(arcsin(2x))+c = frac{1}{16}(arcsin(2x) - x) + c\int {x*arccos(2x)dx} = frac{1}{2}x^2*arccos(2x) + frac{1}{16}(arcsin(2x) - x) + c$

$int{frac{x}{x^3-x^2}dx = int{frac{1}{(x-1)(x+1)}}dx = int{frac{1}{2}(frac{1}{x-1} - frac{1}{x+1})}dx = frac{1}{2}(ln|x-1| - ln|x+1|)+c = frac{1}{2}ln|frac{x-1}{x+1}| + c$$int{frac{13}{(x^2+4)(x+3)}dx = | frac{Ax+B}{x^2+4} + frac{C}{x+3} = frac{(Ax+B)(x+3) + Cx^2 + 4C}{(x+3)(x^2+4)} = frac{Ax^2 + 3Ax+Bx+3B + Cx^2+4C}{(x+3)(x^2+4)} =$$frac{x^2(A+C)+x(3A+B) + 3B + 4C}{(x+3)(x^2+4)} => left { {{A+C=0} atop {3A+B=0\} atop {3B+4C=13}} right. => left { {{A=-C} atop {B = -3A = 3C\} atop {9C+4C=13 => 13C = 13 => C = 1}} right.$$A = -1;\B = 3;\C = 1\frac{13}{(x^2+4)(x+3)} = frac{3-x}{x^2+4} + frac{1}{x+3}|\ int{frac{13}{(x^2+4)(x+3)}dx} = int{ frac{3-x}{x^2+4} + frac{1}{x+3}}dx = -int{frac{x-3}{x^2+4}dx + ln(x+3) = -int{frac{xdx}{x^2+4} + 3intfrac{dx}{x^2+2^2} + ln(x+3)$$-frac{1}{2}ln(x^2+4) + frac{3}{2}arctgfrac{x}{2} + ln(x+3) + c$