Question

To integrate the expression Ax + B /(γx² + r), split it into two fractions. What is the form you want to achieve for integration?

a) k∫f'⁽x⁾/fₓ dx
b) k∫ln(f(x)) dx
c) k∫(Ax/(γx² + r) + B/(γx² + r)) dx
d) k∫(Ax + B)/(γx² + r) dx

Answer 1

The goal of splitting the expression into two fractions is to create a form amenable to integration, with one part resembling the derivative-over-function form and the other as a constant over the denominator.

Step-by-step explanation:

To integrate the expression Ax + B over the denominator (γx² + r), the expression should first be split into two separate fractions where one has a numerator of some multiple of the derivative of the denominator and the other is a constant over the denominator. The ultimate goal is to obtain a form that allows for direct integration, usually aiming to utilize integration techniques such as u-substitution or finding an antiderivative that resembles the natural logarithm.

The desired form would be k∫(Ax/(γx² + r) + B/(γx² + r)) dx, which is essentially the original integrand split into two parts that can each be integrated more easily. Specifically, the first part is aimed to be in the form of derivative-over-function (k∫f'(x)/f(x) dx), which leads to a natural logarithm when integrated, and the second part could potentially be resolved through simpler integration methods if it takes on a more convenient form.