What is int (-2x^3+4x ) / (-2x^2+x +7 )?

1 Answer
Oct 15, 2016

int(2x^3 - 4x)/(2x^2 - x - 7)dx = x^2/2 + x/2 + 7/456 ((57+5 sqrt(57)) ln(-4 x+sqrt(57)+1)+(57-5 sqrt(57)) ln(4 x+sqrt(57)-1))+ C

Explanation:

int(-2x^3 + 4x)/(-2x^2 + x + 7)dx

Multiply by 1 in the form (-1)/-1:

int(2x^3 - 4x)/(2x^2 - x - 7)dx

The power of the numeration is greater than the power of the denominator, therefore, we do the implied division:

........................x + 1/2
2x^2 - x - 7|2x^3 + 0x^2 - 4x + 0
..................-2x^2 + x^2 + 7x
...............................x^2 + 3x
..............................-x^2 +1/2x + 7/2
........................................(7/2)x + 7/2

int(2x^3 - 4x)/(2x^2 - x - 7)dx = intxdx + 1/2intdx + 7/2int(x + 1)/(2x^2 - x -7)dx

intxdx = x^2/2:

int(2x^3 - 4x)/(2x^2 - x - 7)dx = x^2/2 + 1/2intdx + 7/2int(x + 1)/(2x^2 - x -7)dx

1/2intdx = x/2:

int(2x^3 - 4x)/(2x^2 - x - 7)dx = x^2/2 + x/2 + 7/2int(x + 1)/(2x^2 - x -7)dx

Integration of the last integral by wolframalpha

int(2x^3 - 4x)/(2x^2 - x - 7)dx = x^2/2 + x/2 + 7/456 ((57+5 sqrt(57)) ln(-4 x+sqrt(57)+1)+(57-5 sqrt(57)) ln(4 x+sqrt(57)-1))+ C