Evaluate the following integrals :
`int (sin x+cos x+x^(5//2))dx.`

Answer

To evaluate the integral \( \int (\sin x + \cos x + x^{5/2}) \, dx \), we can break it down into separate integrals.

### Step-by-Step Solution:

1. **Separate the Integral:**
\[ \int (\sin x + \cos x + x^{5/2}) \, dx = \int \sin x \, dx + \int \cos x \, dx + \int x^{5/2} \, dx \]

2. **Integrate \( \sin x \):**
The integral of \( \sin x \) is:
\[ \int \sin x \, dx = -\cos x + C_1 \]

3. **Integrate \( \cos x \):**
The integral of \( \cos x \) is:
\[ \int \cos x \, dx = \sin x + C_2 \]

4. **Integrate \( x^{5/2} \):**
Using the power rule for integration, we have:
\[ \int x^{5/2} \, dx = \frac{x^{5/2 + 1}}{5/2 + 1} + C_3 = \frac{x^{7/2}}{7/2} + C_3 = \frac{2}{7} x^{7/2} + C_3 \]

5. **Combine the Results:**
Now, we combine all the results from the integrals:
\[ \int (\sin x + \cos x + x^{5/2}) \, dx = -\cos x + \sin x + \frac{2}{7} x^{7/2} + C \]
where \( C = C_1 + C_2 + C_3 \) is the constant of integration.

6. **Final Answer:**
Thus, the final result is:
\[ \int (\sin x + \cos x + x^{5/2}) \, dx = -\cos x + \sin x + \frac{2}{7} x^{7/2} + C \]