∫(x^2 - 4)/((x + 2)(x - 1)(x - 3)) dx =

The correct answer is Option (3) → \(\displaystyle\frac{1}{2}\log_e\left|C(x - 1)(x - 3)\right|\)

Since \(x^2 - 4 = (x - 2)(x + 2)\), the factor \(x + 2\) cancels (x ≠ −2):

    \( \displaystyle \frac{x^2 - 4}{(x + 2)(x - 1)(x - 3)} = \frac{x - 2}{(x - 1)(x - 3)} \)

    \( \displaystyle = \frac{x - 2}{x^2 - 4x + 3} \)

The derivative of the denominator is \(2x - 4 = 2(x - 2)\), so the numerator is half the derivative of the denominator:

    \( \displaystyle \int\frac{x - 2}{x^2 - 4x + 3}\,dx = \frac{1}{2}\int\frac{2x - 4}{x^2 - 4x + 3}\,dx \)

    \( \displaystyle = \frac{1}{2}\log_e\left|x^2 - 4x + 3\right| + \frac{1}{2}\log_e|C| \)

    \( \displaystyle = \frac{1}{2}\log_e\left|C(x - 1)(x - 3)\right| \)

Here the constant of integration is written as \(\frac{1}{2}\log_e|C|\), which is why C appears inside the logarithm.