Integrating basic equation

Question

In the following question I am just wondering if there is another way to solve this integral without using partial fraction decomposition.

$$\int \frac{1}{x(2-x)}dx$$

If there is another way what is it called?

Answer 1

We can also complete the square. We see $$\frac{1}{x(2-x)}= \frac{1}{2x-x^2}= \frac{1}{1-(x-1)^2}.$$ Thus, the integral becomes

$$\int \frac{1}{1-(x-1)^2} \ dx = \tanh^{-1}(x-1) + C.$$