Which of the following is an anti-derivative of \[ \frac{1}{4x^2 + 1} \] ? A - HSC - SSCE Mathematics Extension 1 - Question 3 - 2020 - Paper 1

The integral formula for ( \int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + c ) suggests that we can rewrite our integrand as follows:

Given ( 4x^2 + 1 = 1^2 + (2x)^2 ), we identify ( a = 1 ) and ( x = 2x ).

Thus, we can apply the formula with:

• ( a = 2 )
• ( x = 2x )

Applying the formula, we obtain:

[ \int \frac{1}{4x^2 + 1} dx = \frac{1}{2} \tan^{-1}(2x) + c ]

This implies that the correct anti-derivative is given in option D.

The correct answer is therefore:

D. ( \frac{1}{2} \tan^{-1}(2x) + c )