The correct option is A 2log|x2+3x+2|−5log∣∣∣x+1x+2∣∣∣+c
From the previous question we know,4x+1 x2+3x+2=2(2x+3)−5x2+3x+2
Now, we can express the integral as
I=∫2(2x+3)−5 dxx2+3x+2
=2∫(2x+3) dxx2+3x+2−5∫dxx2+3x+2
=2log(x2+3x+2∣∣−5∫dxx2+3x+(9/4)−(9/4)+2 + c
=2log(x2+3x+2∣∣−5∫dx(x+3/2)2−(1/2)2 + c
=2log(x2+3x+2∣∣−512(1/2)log(x+32−12x+32+12∣∣
∣∣ + c
=2log(x2+3x+2∣∣−5log(x+1x+2∣∣+c