\( \int 3 \cos ^ { 2 } ( x ) \tan ^ { 3 } ( x ) d x \)

To solve the integral \( \int 3 \cos^{2}(x) \tan^{3}(x) \, dx \), we can first utilize the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) to express everything in terms of sine and cosine. Notably, \( \tan^3(x) = \frac{\sin^3(x)}{\cos^3(x)} \), and thus the integral becomes: \[ \int 3 \cos^{2}(x) \frac{\sin^{3}(x)}{\cos^{3}(x)} \, dx = \int 3 \frac{\sin^{3}(x)}{\cos(x)} \, dx \] To solve this integral, we can perform a substitution. Let \( u = \sin(x) \), which gives \( du = \cos(x) \, dx \). The integral now transforms to \[ 3 \int \frac{u^3}{\sqrt{1 - u^2}} \, du \] This integral may require further techniques such as integration by parts or trigonometric substitution depending on its complexity. If the methods get tricky, looking into integration tables could be beneficial! For a real-world application, integrals like this are commonly used in physics to calculate work done along a path or to analyze waves in mechanics. Understanding these calculations can help in modeling real-world systems dynamically and effectively, bringing practical insights from calculus into fields like engineering and physics!