∫2x/(x²-2x+6)dx
=∫[2(x-1)+2]/[4+(x-1)²]dx
=2∫[(x-1)/[4+(x-1)²]dx+2∫1/[4+(x-1)²]dx
=2∫[(x-1)/[4+(x-1)²]d(x-1)+2∫1/[4+(x-1)²]d(x-1)
=∫[1/[4+(x-1)²]d(x-1)²+2∫1/[4+(x-1)²]d(x-1)
=∫[1/[4+(x-1)²]d[4+(x-1)²]+2∫1/[2²+(x-1)²]d(x-1)
=ln[4+(x-1)²]+arctan[(x-1)/2]+C