Which expression is equal to \[ \int \frac{1}{x^2 + 4x + 10} \, dx \] ? A - HSC - SSCE Mathematics Extension 2 - Question 6 - 2020 - Paper 1

Next, we'll perform a substitution. Let: [ u = x + 2, ] in which case, ( dx = du ), and the integral becomes: [ \int \frac{1}{u^2 + 6} , du. ] This integral can be solved using the formula: [ \int \frac{1}{a^2 + u^2} , du = \frac{1}{a} \tan^{-1}\left( \frac{u}{a} \right) + c. ] Here, ( a = \sqrt{6} ). So the integral simplifies to: [ \frac{1}{\sqrt{6}} \tan^{-1}\left( \frac{u}{\sqrt{6}} \right) + c. ]

Replacing ( u ) with ( x + 2 ), we have: [ \frac{1}{\sqrt{6}} \tan^{-1}\left( \frac{x + 2}{\sqrt{6}} \right) + c. ] Thus, the answer corresponds to option A.