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

Now we substitute ( u = x + 2 ), leading to:

[ du = dx ] Thus, the integral transforms into:

[ \int \frac{1}{u^2 + 6} ; du ]

The integral ( \int \frac{1}{u^2 + a^2} ; du = \frac{1}{a} \tan^{-1}\left( \frac{u}{a} \right) + c ) applies here with ( a = \sqrt{6} ):

[ \int \frac{1}{u^2 + 6} ; du = \frac{1}{\sqrt{6}} \tan^{-1}\left( \frac{u}{\sqrt{6}} \right) + c ]

Returning to original variable, we substitute back ( u = x + 2 ):

[ \frac{1}{\sqrt{6}} \tan^{-1}\left( \frac{x + 2}{\sqrt{6}} \right) + c ] This corresponds to option A in the original question.