By introducing a substitution,
t=3(x−1)2−12;x=t+123+1;dx=dt62t+1;∫dx3x2−6x+2=16∗2∫dtt+12t−12=112∫dtt2−14=112log∣t+t2−14∣=123log∣3(x−1)2−12+(x−1)3(3x2−6x+2)∣.t = 3(x-1)^2 - \frac{1}{2}; \\ x = \sqrt{\frac{t+\frac{1}{2}}{3}}+1; \\ dx = \frac{dt}{\sqrt{6}\sqrt{2t+1}}; \\ \smallint{\frac{dx}{\sqrt{3x^2-6x+2}}} = \frac{1}{\sqrt{6*2}}\smallint{\frac{dt}{\sqrt{t+\frac{1}{2}}\sqrt{t-\frac{1}{2}}}} = \frac{1}{\sqrt{12}}\smallint{\frac{dt}{\sqrt{t^2-\frac{1}{4}}}} = \frac{1}{\sqrt{12}}\log{|t+\sqrt{t^2-\frac{1}{4}}|} = \frac{1}{2\sqrt{3}}\log{|3(x-1)^2 - \frac{1}{2}+(x-1)\sqrt{3(3x^2-6x+2)}|}.t=3(x−1)2−21;x=3t+21+1;dx=62t+1dt;∫3x2−6x+2dx=6∗21∫t+21t−21dt=121∫t2−41dt=121log∣t+t2−41∣=231log∣3(x−1)2−21+(x−1)3(3x2−6x+2)∣.
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