Evaluate the following :
`int (cos x+cosec^2 x-x^8+1) dx.`

Answer

To evaluate the integral \( \int (\cos x + \csc^2 x - x^8 + 1) \, dx \), we will break it down into individual components and integrate each term separately.

### Step-by-Step Solution:

1. **Identify the integral**:
\[ \int (\cos x + \csc^2 x - x^8 + 1) \, dx \]

2. **Integrate each term separately**:
- **First term**: \( \int \cos x \, dx \)
- The integral of \( \cos x \) is \( \sin x \).
- **Second term**: \( \int \csc^2 x \, dx \)
- The integral of \( \csc^2 x \) is \( -\cot x \).
- **Third term**: \( \int -x^8 \, dx \)
- The integral of \( -x^8 \) is \( -\frac{x^9}{9} \).
- **Fourth term**: \( \int 1 \, dx \)
- The integral of \( 1 \) is \( x \).

3. **Combine the results**:
Putting all the integrated terms together, we have:
\[ \sin x - \cot x - \frac{x^9}{9} + x + C \]
where \( C \) is the constant of integration.

### Final Answer:
\[ \int (\cos x + \csc^2 x - x^8 + 1) \, dx = \sin x - \cot x - \frac{x^9}{9} + x + C \]