HSC Maths Advanced Syllabus

The Reciprocal Function

As we know the reciprocal function is displayed by $\frac{d}{dx}log_ex=\frac{1}{x}$. By reversing this we can determine the formula to integrate expressions in the form $\frac{1}{x}$. Thus, the formula to integrate the reciprocal function, is as follows:

$$\int\frac{1}{x}dx=log_ex+C$$

However, the function $\frac{1}{x}$ can also be negative, and as such, has primitive $log_e(-x)$ as well as $log_ex$. Thus, we manipulate the formula for integrating the reciprocal function to be:

$$\int\frac{1}{x}dx=log_e|x|+C$$

Additionally, there are two other standard formulas used for integrating the reciprocal function.

$$\int\frac{1}{ax+b}dx=\frac{1}a{}log_e|ax+b|+C$$ $$\int\frac{u'}{u}dx=log_e|u|+C \quad or \quad \int\frac{f'(x)}{f(x)}dx=log_e|f(x)|+C$$

When we are asked to solve increasingly difficult integration questions, the process in which we we solve the question can vary slightly. In the video above, we explore this concept, looking at solving the integral: $\int_1^elog_exdx$