Evaluate the following integrals:
`int_(a)^(b)f(x)dx +int_(b)^(a)f(x)dx`

Answer

To evaluate the integral \( \int_{a}^{b} f(x) \, dx + \int_{b}^{a} f(x) \, dx \), we can use the properties of definite integrals.

### Step-by-Step Solution:

1. **Understanding the Integral**:
We have two integrals:
- The first integral is from \( a \) to \( b \): \( \int_{a}^{b} f(x) \, dx \).
- The second integral is from \( b \) to \( a \): \( \int_{b}^{a} f(x) \, dx \).

2. **Using the Property of Definite Integrals**:
The property of definite integrals states that:
\[ \int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx \]
This means that reversing the limits of integration changes the sign of the integral.

3. **Substituting the Property**:
Now we can substitute this property into our original expression:
\[ \int_{a}^{b} f(x) \, dx + \int_{b}^{a} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \left(-\int_{a}^{b} f(x) \, dx\right) \]

4. **Simplifying the Expression**:
This simplifies to:
\[ \int_{a}^{b} f(x) \, dx - \int_{a}^{b} f(x) \, dx = 0 \]

5. **Final Result**:
Therefore, the value of the integral \( \int_{a}^{b} f(x) \, dx + \int_{b}^{a} f(x) \, dx \) is:
\[ \boxed{0} \]