\( \int _{}^{}\frac{1}{x\sqrt {x-1}} d \)

To solve the integral \( \int \frac{1}{x\sqrt{x-1}} \, dx \), a popular technique is to use a trigonometric substitution. By letting \( x - 1 = \tan^2(\theta) \), we derive that \( \sqrt{x - 1} = \tan(\theta) \) and \( dx = 2\tan(\theta)\sec^2(\theta)d\theta \). This transforms our integral into a form that is easier to manage, eventually leading us to express it in terms of a more familiar integral or a known function. Upon solving with this substitution, you'll get a new integral or expression in terms of \( \theta \), and after integrating, don’t forget to revert back to the variable \( x \) using the relationship you derived initially. It’s a fun process of transforming complex integrals into solvable forms! Don’t forget that trigonometric identities play a crucial role here, so it’s helpful to be familiar with those as you wind your way through the calculations. And be careful with each step—small mistakes in substitution can lead to significant confusion later on in solving! Happy integrating!