Absolute value's antiderivative plays hide-and-seek. It splits at zero, mimicking x on the positive side and -x on the negative, each with their own +C companion. A two-faced dance, bound by the constant's hand.
The absolute value function, |x|, throws a curveball at antiderivative seekers. Unlike most functions, it doesn't offer a single, smooth ride across its domain. Instead, it has a "split personality," behaving differently on either side of zero.
For the sunny side, x ≥ 0, the journey is familiar. Here, |x| embraces its true identity as x, and its antiderivative unfolds as the classic x²/2 + C, where C is the ever-present constant companion in integration.
But step into the shadows, x < 0, and the function undergoes a Jekyll-and-Hyde transformation. Here, |x| sheds its x disguise and dons a negative mask, becoming -x. Its antiderivative reflects this shift, taking the form -x²/2 + C.
So, while the absolute value function might seem singular, its antiderivative tells a tale of two paths. We must navigate each side separately, acknowledging their distinct identities and antiderivative expressions, both adorned with the ever-present C.
In essence, the absolute value function's antiderivative dances a two-step, with each side following its own rhythm, forever linked by the constant beat of C.
This hopefully provides a more expansive and informative picture of the antiderivative landscape for the absolute value function, without repeating the same points.