Final answer:
To integrate the square root of x² - 16 using trigonometric identities, we can perform a substitution such as x = 4 sec(θ), which transforms the integral into a trigonometric one solvable using standard techniques.
Step-by-step explanation:
The integration of the root of x² - 16 can be solved using trigonometric substitution. Trigonometric identities can be particularly useful in integrals involving square roots of expressions such as x² - a², which suggests using a substitution like x = a sec(θ) for a² + x² type expressions or x = a cos(θ) for a² - x² type expressions.
When integrating square root functions with expressions like x² - 16, we can use a substitution like x = 4 sec(θ) considering the identity sec²(θ) - 1 = tan²(θ), transforming the integral into a trigonometric form that can be integrated using basic integration rules.
Here's a step-by-step approach to this type of integral: First, we make a substitution x = 4 sec(θ). This gives us √(x² - 16) = √(16 sec²(θ) - 16) = 4 tan(θ). Then, we find dx which is 4 sec(θ) tan(θ) dθ. The integral now becomes an integral in terms of θ, which we can solve using basic trigonometric integrals.