What is int (-3x^3+4 ) / (-2x^2+x +1 )∫−3x3+4−2x2+x+1?

1 Answer
Cem Sentin
Nov 22, 2017

int ((-3x^3+4)*dx)/(-2x^2+x+1)∫(−3x3+4)⋅dx−2x2+x+1

=(3x^2+3x)/4+35/24*Ln(2x+1)-1/3*Ln(x-1)+C3x2+3x4+3524⋅ln(2x+1)−13⋅ln(x−1)+C

Explanation:

int ((-3x^3+4)*dx)/(-2x^2+x+1)∫(−3x3+4)⋅dx−2x2+x+1

=int ((3x^3-4)*dx)/(2x^2-x-1)∫(3x3−4)⋅dx2x2−x−1

=1/4*int ((12x^3-16)*dx)/(2x^2-x-1)14⋅∫(12x3−16)⋅dx2x2−x−1

=1/4*int ([(2x^2-x-1)*(6x+3)+9x-13]*dx)/(2x^2-x-1)14⋅∫[(2x2−x−1)⋅(6x+3)+9x−13]⋅dx2x2−x−1

=1/4*int (6x+3)*dx+1/4*int ((9x-13)*dx)/(2x^2-x-1)14⋅∫(6x+3)⋅dx+14⋅∫(9x−13)⋅dx2x2−x−1

=(3x^2+3x)/43x2+3x4+1/4*int ((9x-13)dx)/[(2x+1)(x-1)]14⋅∫(9x−13)dx(2x+1)(x−1)

=(3x^2+3x)/4+1/4*[35/3*int dx/(2x+1)-4/3*int dx/(x-1)]3x2+3x4+14⋅[353⋅∫dx2x+1−43⋅∫dxx−1]

=(3x^2+3x)/4+35/12*int dx/(2x+1)-1/3*int dx/(x-1)3x2+3x4+3512⋅∫dx2x+1−13⋅∫dxx−1

=(3x^2+3x)/4+35/24*Ln(2x+1)-1/3*Ln(x-1)+C3x2+3x4+3524⋅ln(2x+1)−13⋅ln(x−1)+C

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