The tutor shows the example ∫dx/(x²+6)

∫dx/(x²+1) = arctanx + C

The related integral

∫dx/(x²+6)

must be put in the form, as follows:

∫dx/(x²+6) = ∫dx/(6(x²/6+1)) = 1/6 ∫dx/(x²/6 + 1)

=1/6 ∫dx/((x/√6)²+1) = (√6)/6∫(dx(1/√6))/((x/√6)² + 1)

Next it becomes

1/√6∫(dx(1/√6))/((x/√6)² + 1)

which can be integrated:

1/√6 ∫(dx(1/√6))/((x/√6)² + 1)=(1/√6)arctan(x/√6) + C

HTH:)

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus, 3rd ed. Toronto: D C Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.