`int sec^(2//3) x cosec^(4//3) x dx` is equal to

Options

A. `3(tan x)^(1//3)`
B. `3(cot x)^(-1//3)`
C. `-3(tan x)^(-1//3)`
D. `-3(cot x)^(-1//3)`

Answer

To solve the integral \( \int \sec^{\frac{2}{3}} x \csc^{\frac{4}{3}} x \, dx \), we can follow these steps:

### Step 1: Convert to Sine and Cosine
We start by rewriting the integral in terms of sine and cosine:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x} \]
Thus, we can rewrite the integral as:
\[ \int \sec^{\frac{2}{3}} x \csc^{\frac{4}{3}} x \, dx = \int \left(\frac{1}{\cos x}\right)^{\frac{2}{3}} \left(\frac{1}{\sin x}\right)^{\frac{4}{3}} \, dx = \int \frac{1}{\cos^{\frac{2}{3}} x \sin^{\frac{4}{3}} x} \, dx \]

### Step 2: Rewrite the Integral
We can express the integral as:
\[ \int \frac{\sin^{\frac{4}{3}} x}{\cos^{\frac{2}{3}} x} \, dx = \int \sec^{\frac{2}{3}} x \sin^{\frac{4}{3}} x \, dx \]
This can be rewritten as:
\[ \int \sec^{2} x \cdot \sec^{-\frac{4}{3}} x \sin^{\frac{4}{3}} x \, dx \]

### Step 3: Use Substitution
Let us set \( t = \tan x \). Then, we have:
\[ \frac{dt}{dx} = \sec^2 x \quad \Rightarrow \quad dx = \frac{dt}{\sec^2 x} \]
Substituting this into the integral gives:
\[ \int \sec^{-\frac{4}{3}} x \sin^{\frac{4}{3}} x \cdot \frac{dt}{\sec^2 x} \]
This simplifies to:
\[ \int t^{\frac{4}{3}} \cdot \sec^{-\frac{4}{3}} x \cdot \frac{dt}{\sec^2 x} \]

### Step 4: Solve the Integral
Now, we can express the integral in terms of \( t \):
\[ \int t^{\frac{4}{3}} \cdot t^{-\frac{4}{3}} \, dt = \int t^{-\frac{4}{3}} \, dt \]
The integral of \( t^{-\frac{4}{3}} \) is given by:
\[ \int t^{-\frac{4}{3}} \, dt = \frac{t^{-\frac{1}{3}}}{-\frac{1}{3}} = -3t^{-\frac{1}{3}} + C \]

### Step 5: Substitute Back
Now, substituting back \( t = \tan x \):
\[ -3 \tan^{-\frac{1}{3}} x + C \]
This can be rewritten as:
\[ -3 \cdot \frac{1}{\tan^{\frac{1}{3}} x} + C = -3 \cdot \cot^{\frac{1}{3}} x + C \]

### Final Answer
Thus, the final result of the integral is:
\[ -3 \cot^{\frac{1}{3}} x + C \]