Use substitution to evaluate the trigonometric indefinite integral
To evaluate an indefinite integral that involves trigonometric functions, a common technique is to use substitution, often called "u-substitution." The idea behind substitution is to simplify the integrand (the function being integrated) by identifying a part of it as a new variable, usually denoted as "u". This helps transform the integral into a form that is easier to work with. For integrals involving inverse trigonometric functions, like arcsine, it's common to recognize patterns from known derivatives of trig functions and their inverses. You look for a part of the integrand that matches the derivative of something else, and then use this to replace variables.
In this specific type of problem, you might recognize that the expression inside the square root is related to the derivative of arcsine. By setting "u" equal to a trigonometric expression (like arcsine or a function of x), you can then rewrite the integral in terms of u. This transforms the integral into something more straightforward, which you can then solve using basic integration rules. After solving, you substitute back the original variable to get the final answer. This technique helps break down more complex trigonometric integrals into simpler forms that are easier to evaluate.