Evaluate the following integrals :
`int (3 cosec^2 x-5x+sinx ) dx.`

Answer

To evaluate the integral \( \int (3 \csc^2 x - 5x + \sin x) \, dx \), we will break it down into simpler parts using the linearity property of integrals.

### Step-by-Step Solution:

1. **Split the Integral**:
\[ \int (3 \csc^2 x - 5x + \sin x) \, dx = \int 3 \csc^2 x \, dx - \int 5x \, dx + \int \sin x \, dx \]

2. **Evaluate \( \int 3 \csc^2 x \, dx \)**:
- We know that \( \int \csc^2 x \, dx = -\cot x + C \).
- Therefore:
\[ \int 3 \csc^2 x \, dx = 3 \left(-\cot x\right) = -3 \cot x \]

3. **Evaluate \( \int 5x \, dx \)**:
- Using the power rule, \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \):
\[ \int 5x \, dx = 5 \cdot \frac{x^2}{2} = \frac{5}{2} x^2 \]

4. **Evaluate \( \int \sin x \, dx \)**:
- We know that \( \int \sin x \, dx = -\cos x + C \):
\[ \int \sin x \, dx = -\cos x \]

5. **Combine All Parts**:
- Now, we combine all the evaluated integrals:
\[ \int (3 \csc^2 x - 5x + \sin x) \, dx = -3 \cot x - \frac{5}{2} x^2 - \cos x + C \]

### Final Answer:
\[ \int (3 \csc^2 x - 5x + \sin x) \, dx = -3 \cot x - \frac{5}{2} x^2 - \cos x + C \]