∫21dxx2+5x+6
=∫21dxx2+2x+3x+6=∫21dxx(x+2)+3(x+2)
=∫21dx(x+3)(x+2)
=∫21(x+3)−(x+2)(x+3)(x+2)dx [∵(x+3)−(x+2)=1]
=∫21x+3(x+3)(x+2)−(x+2)(x+3)(x+2)dx
=∫211x+2−1x+3dx
=[log|x+2|−log|x+3|]21
=log|x+2||x+3|]21 [∵loga−logb=logab]
=log∣∣2+22+3∣∣−log∣∣1+21+3∣∣
=log∣∣45∣∣−log∣∣34∣∣=log∣∣45×43∣∣=log∣∣1615∣∣
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