If `d/dx (ϕ(x))=f(x)`, then`int_(1)^(2) f(x) dx` is ______

Explanation

To solve the given problem, we need to find the value of the definite integral \( \int_{1}^{2} f(x) \, dx \) given that \( \frac{d}{dx} \phi(x) = f(x) \).

### Step-by-step Solution:

1. **Understand the Given Information**:
We are given that \( \frac{d}{dx} \phi(x) = f(x) \). This means that \( f(x) \) is the derivative of the function \( \phi(x) \).

2. **Set Up the Integral**:
We need to evaluate the integral \( \int_{1}^{2} f(x) \, dx \). Since we know that \( f(x) = \frac{d}{dx} \phi(x) \), we can substitute this into the integral:
\[ \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \frac{d}{dx} \phi(x) \, dx \]

3. **Apply the Fundamental Theorem of Calculus**:
According to the Fundamental Theorem of Calculus, if \( F(x) \) is an antiderivative of \( f(x) \), then:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
In our case, \( \phi(x) \) is an antiderivative of \( f(x) \). Therefore, we can write:
\[ \int_{1}^{2} \frac{d}{dx} \phi(x) \, dx = \phi(2) - \phi(1) \]

4. **Final Result**:
Thus, the value of the integral \( \int_{1}^{2} f(x) \, dx \) is:
\[ \phi(2) - \phi(1) \]

### Conclusion:
The final answer is \( \phi(2) - \phi(1) \).

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