Step 1: Understanding the Concept:
This integral involves fractional powers of trigonometric functions. The goal is to express the integrand in terms of $\tan x$ and its derivative $\sec^2 x$. Step 2: Key Formula or Approach:
Rewrite the expression in terms of $\sin x$ and $\cos x$:
\[ I = \int \frac{1}{\cos^{\frac{2}{3}} x \cdot \sin^{\frac{4}{3}} x} dx \]
Then, divide the numerator and denominator by $\cos^2 x$ to introduce $\tan x$. Step 3: Detailed Explanation:
\[ I = \int \frac{\sec^2 x}{\tan^{\frac{4}{3}} x} dx \]
Let $t = \tan x \implies dt = \sec^2 x dx$.
Substituting into the integral:
\[ I = \int t^{-\frac{4}{3}} dt \]
Using the power rule for integration $\int t^n dt = \frac{t^{n+1}}{n+1}$:
\[ I = \frac{t^{-\frac{4}{3} + 1}}{-\frac{4}{3} + 1} + c = \frac{t^{-\frac{1}{3}}}{-\frac{1}{3}} + c \]
\[ I = -3 t^{-\frac{1}{3}} + c = -3 \tan^{-\frac{1}{3}} x + c \]
Step 4: Final Answer:
The integral is $-3 \tan^{\frac{-1}{3}} x + c$.