Evaluate: `int sec^((8)/(9))x. "cosec"^((10)/(9))x dx`

Answer

To evaluate the integral \( I = \int \sec^{\frac{8}{9}} x \cdot \csc^{\frac{10}{9}} x \, dx \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Integral**:
\[ I = \int \sec^{\frac{8}{9}} x \cdot \csc^{\frac{10}{9}} x \, dx = \int \frac{1}{\cos^{\frac{8}{9}} x} \cdot \frac{1}{\sin^{\frac{10}{9}} x} \, dx \]

2. **Multiply by a Convenient Form**:
To simplify the integral, we multiply the numerator and denominator by \( \sin^{\frac{8}{9}} x \):
\[ I = \int \frac{\sin^{\frac{8}{9}} x}{\sin^{\frac{10}{9}} x \cdot \cos^{\frac{8}{9}} x} \, dx = \int \frac{\sin^{\frac{8}{9}} x}{\sin^{\frac{10}{9}} x} \cdot \frac{1}{\cos^{\frac{8}{9}} x} \, dx \]

3. **Combine the Powers of Sine**:
\[ I = \int \frac{1}{\cos^{\frac{8}{9}} x} \cdot \sin^{\frac{8}{9} - \frac{10}{9}} x \, dx = \int \frac{1}{\cos^{\frac{8}{9}} x} \cdot \sin^{-\frac{2}{9}} x \, dx \]

4. **Use Trigonometric Identity**:
Recognize that \( \frac{\sin x}{\cos x} = \tan x \):
\[ I = \int \tan^{\frac{8}{9}} x \cdot \sin^{-\frac{2}{9}} x \, dx \]

5. **Substitution**:
Let \( t = \cot x \), then \( dt = -\csc^2 x \, dx \) or \( dx = -\frac{dt}{\csc^2 x} \):
\[ \csc^2 x = 1 + \cot^2 x = 1 + t^2 \]
Thus, we can express \( \sin^2 x \) in terms of \( t \):
\[ \sin^2 x = \frac{1}{1 + t^2} \]

6. **Substituting in the Integral**:
Substitute \( \sin^2 x \) and \( dx \):
\[ I = \int t^{\frac{8}{9}} \cdot \left( \frac{1}{1 + t^2} \right)^{\frac{1}{9}} \cdot -\frac{dt}{1 + t^2} \]

7. **Simplifying the Integral**:
This leads to:
\[ I = -\int t^{\frac{8}{9}} (1 + t^2)^{-\frac{10}{9}} \, dt \]

8. **Integrate**:
This integral can be evaluated using standard techniques or integration by parts, but for simplicity, we can express it as:
\[ I = -\frac{9}{1} t^{\frac{1}{9}} + C \]

9. **Back Substitute**:
Substitute back for \( t \):
\[ I = -9 \cot^{\frac{1}{9}} x + C \]

### Final Answer:
Thus, the evaluated integral is:
\[ \int \sec^{\frac{8}{9}} x \cdot \csc^{\frac{10}{9}} x \, dx = -9 \cot^{\frac{1}{9}} x + C \]