Calculus 2: Trigonometric substitution

Identify the type of trigonometric substitution needed (sine, tangent, secant) to integrate a function involving expressions such as $a^2 - x^2$, $a^2 + x^2$, or $x^2 - a^2$, complete the necessary substitutions, and integrate the resulting expression.

Evaluate the definite integral $\displaystyle \int_{4}^{6} \frac{x^2}{\sqrt{x^2 - 9}} \, dx$ by using trigonometric substitution in two different ways: first by computing the antiderivative and then using the Fundamental Theorem of Calculus with the original limits of integration; second by changing the limits of integration after substituting $x = 3 \sec(\theta)$ and evaluating the integral in terms of $\theta$.

Solve $\displaystyle \int \frac{\sec \theta \cdot 4 \tan^2 \theta}{\cos^2 \theta} \, d\theta$ using a substitution method.