Integral((x-3)dx/(3 - 2x - x^3))
(3 - 2x - x^3) = 0
подставим х = 1
получим 3 - 2*1 - 1^3 = 3 - 2 - 1 = 3 - 3 = 0
поэтому (x - 1) - один из элементов разложения полинома (3 - 2x - x^3)
(3 - 2x - x^3) =
-(x^3 + 2x - 3) =
-(x^3 - x + 3х - 3) =
-(x(х^2 - 1) + 3(х - 1)) =
-(x(х^2 - 1) + 3(х - 1)) =
-(х(х - 1)(х + 1) + 3(х - 1)) =
-(х - 1)(х(х + 1) + 3) =
-(х - 1)(х^2 + x + 3) =
-(х - 1)(х^2 + x + 3)
(x-3)/(3 - 2x - x^3) =
-(x - 3)/((х - 1)(х^2 + x + 3)) =
-(x - 1 - 2)/((х - 1)(х^2 + x + 3)) =
-(x - 1)/((х - 1)(х^2 + x + 3)) + 2/((х - 1)(х^2 + x + 3)) =
-1/(х^2 + x + 3) + 2/((х - 1)(х^2 + x + 3)) =
-1/(х^2 + x + 3) + 2/(x^3 + 2x - 3)
integral((x-3)dx/(3 - 2x - x^3)) =
integral(-1/(х^2 + x + 3) + 2/(x^3 + 2x - 3))dx =
integral(-1/(х^2 + x + 3))dx + integral(2/(x^3 + 2x - 3))dx
integral(-1/(х^2 + x + 3))dx =
-integral(1/(х^2 + x + 3))dx =
-integral(1/(х^2 + 2(1/2)x + 1/4 + 11/4))dx =
-integral(1/((х + 1/2)^2 + 11/4))dx =
-(4/11)*integral(1/(( (2/корень(11)) * (х + 1/2) )^2 + 1))dx =
-(2/корень(11))*integral(1/(( (2/корень(11)) * (х + 1/2) )^2 + 1))d(2/корень(11)) * (х + 1/2)) =
-(2/корень(11))*integral(1/(( (2/корень(11)) * (х + 1/2) )^2 + 1))d( (2/корень(11))* (х + 1/2) ) =
- arctg( (2/корень(11))*(х + 1/2) ) + C1
integral(2/(x^3 + 2x - 3))dx =
integral( 2/((х - 1)(х^2 + x + 3)) )dx =
2* integral( 1/((х - 1)(х^2 + x + 3)) )dx
1/((х - 1)(х^2 + x + 3)) = A/(х - 1) + (Bx + C)/(х^2 + x + 3)
1 = A(х^2 + x + 3) + (Bx + C)(х - 1)
1 = Aх^2 + Ax + 3A + Bx^2 + Cх - Bx - C
1 = (A + B)х^2 + (A + C - B)x + (3A - C)
A + B = 0 -> A = -B
A + C - B = 0 -> -B + C - B = 0 -> C -2B = 0 -> C = 2B
3A - C = 1 -> 3(-B) - 2B = 1 -> -3B - 2B = 1 -> -5B = 1 -> B = -1/5
B = -1/5
A = -(-1/5) = 1/5
C = 2*(-1/5) = -2/5
2* integral( 1/((х - 1)(х^2 + x + 3)) )dx =
2*integral( (1/5)/(х - 1) + ( ((-1/5)x+ (-2/5))/(х^2 + x + 3) ) )dx =
(2/5)*integral( 1/(х - 1) + ( (-x - 2)/(х^2 + x + 3) ) )dx =
(2/5)*integral( 1/(х - 1) - ( (x + 2)/(х^2 + x + 3) ) )dx =
(2/5)*integral( 1/(х - 1) )dx - (2/5)*integral( (x + 2)/(х^2 + x + 3) )dx =
(2/5)*ln|x-1| - (2/5)*integral( (x + 2)/(х^2 + x + 3) )dx
integral( (x + 2)/(х^2 + x + 3) )dx =
(1/2)*integral( 2(x + 2)/(х^2 + x + 3) )dx =
(1/2)*integral( (2x + 4)/(х^2 + x + 3) )dx =
(1/2)*integral( (2x + 1 + 3)/(х^2 + x + 3) )dx =
(1/2)*integral( (2x + 1)/(х^2 + x + 3) )dx + (1/2)*integral( 3/(х^2 + x + 3) )dx =
(1/2)*integral( (2x + 1)/(х^2 + x + 3) )dx + (3/2)*integral( 1/(х^2 + x + 3) )dx =
(1/2)*integral( 1/(х^2 + x + 3) )d(х^2 + x + 3) + (3/2)*integral( 1/(х^2 + x + 3) )dx =
(1/2)*ln|х^2 + x + 3| + (3/2)*integral( 1/(х^2 + x + 3) )dx =
(1/2)*ln(х^2 + x + 3) + (3/2)*integral( 1/(х^2 + x + 3) )dx =
(1/2)*ln(х^2 + x + 3) + (3/2)*arctg( (2/корень(11))*(х + 1/2) ) + C2
integral(-1/(х^2 + x + 3))dx + integral(2/(x^3 + 2x - 3))dx =
- arctg( (2/корень(11))*(х + 1/2) ) + C1 + (2/5)*ln|x-1| - (2/5)*integral( (x + 2)/(х^2 + x + 3) )dx =
- arctg( (2/корень(11))*(х + 1/2) ) + C1 + (2/5)*ln|x-1| - (2/5)*((1/2)*ln(х^2 + x + 3) + (3/2)*arctg( (2/корень(11))*(х + 1/2) ) + C2) =
- arctg( (2/корень(11))*(х + 1/2) ) + (2/5)*ln|x-1| - (1/5)*ln(х^2 + x + 3) + (3/5)*arctg( (2/корень(11))*(х + 1/2) ) + C1 - (2/5)*C2 =
- arctg( (2/корень(11))*(х + 1/2) ) + (2/5)*ln|x-1| - (1/5)*ln(х^2 + x + 3) + (3/5)*arctg( (2/корень(11))*(х + 1/2) ) + C =
(2/5)*ln|x-1| - (1/5)*ln(х^2 + x + 3) - (3/5)*arctg( (2/корень(11))*(х + 1/2) ) + C