Given: int (-2x^3-x)/(-4x^2+2x+3) dx
Multiply the integral by (-1)/-1:
int (2x^3 + x)/(4x^2-2x-3)dx
Perform the implied division:
color(white)( (4x^2-2x-3)/color(black)(4x^2-2x-3))(1/2x+1/4color(white)(x+0))/(")" color(white)(x)2x^3+ 0x^2 + x+0)
color(white)(".......................")ul(-2x^3+x^2+3/2x)
color(white)(".....................................")x^2+5/2x
color(white)("..................................")ul(-x^2+1/2x+3/4)
color(white)("...............................................")3x+3/4
int 1/2x+ 1/4+ (3x+3/4)/(4x^2-2x-3)dx
Separate into 3 integrals:
1/2intxdx+ 1/4intdx+ 3int(x+1/4)/(4x^2-2x-3)dx
Integrate the first two integrals:
1/4x^2+ 1/4x+ 3int(x+1/4)/(4x^2-2x-3)dx
Multiply the remaining integral by 8/8:
1/4x^2+ 1/4x+ 3/8int(8x+2)/(4x^2-2x-3)dx
Add to the numerator in the form of -2+2:
1/4x^2+ 1/4x+ 3/8int(8x-2+4)/(4x^2-2x-3)dx
Separate into two fractions:
1/4x^2+ 1/4x+ 3/8int(8x-2)/(4x^2-2x-3)+4/(4x^2-2x-3)dx
Separate into two integrals:
1/4x^2+ 1/4x+ 3/8int(8x-2)/(4x^2-2x-3)dx+3/8int4/(4x^2-2x-3)dx
The first integral is the natural logarithm:
1/4x^2+ 1/4x+ 3/8ln|4x^2-2x-3|+3/2int1/(4x^2-2x-3)dx
The last integral is done by completing the square:
1/4x^2+ 1/4x+ 3/8ln|4x^2-2x-3|+(3sqrt13)/52ln|-4x+sqrt13+1|-(3sqrt13)/52ln|4x+sqrt13-1|+ C
