Evaluate `int_0^2 ((x^2)/(4) - 7x + 5)` dx

Answer

To evaluate the integral \( \int_0^2 \left( \frac{x^2}{4} - 7x + 5 \right) dx \), we will follow these steps:

### Step 1: Set up the integral
We start by writing the integral we need to evaluate:
\[ I = \int_0^2 \left( \frac{x^2}{4} - 7x + 5 \right) dx \]

### Step 2: Break down the integral
We can separate the integral into three parts:
\[ I = \int_0^2 \frac{x^2}{4} \, dx - \int_0^2 7x \, dx + \int_0^2 5 \, dx \]

### Step 3: Evaluate each integral separately
1. **First Integral**:
\[ \int_0^2 \frac{x^2}{4} \, dx = \frac{1}{4} \int_0^2 x^2 \, dx \]
Using the formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \):
\[ \int x^2 \, dx = \frac{x^3}{3} \quad \text{thus,} \quad \int_0^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{2^3}{3} - 0 = \frac{8}{3} \]
Therefore,
\[ \int_0^2 \frac{x^2}{4} \, dx = \frac{1}{4} \cdot \frac{8}{3} = \frac{2}{3} \]

2. **Second Integral**:
\[ \int_0^2 7x \, dx = 7 \int_0^2 x \, dx \]
Using the formula:
\[ \int x \, dx = \frac{x^2}{2} \quad \text{thus,} \quad \int_0^2 x \, dx = \left[ \frac{x^2}{2} \right]_0^2 = \frac{2^2}{2} - 0 = 2 \]
Therefore,
\[ \int_0^2 7x \, dx = 7 \cdot 2 = 14 \]

3. **Third Integral**:
\[ \int_0^2 5 \, dx = 5 \int_0^2 1 \, dx = 5 \cdot [x]_0^2 = 5 \cdot (2 - 0) = 10 \]

### Step 4: Combine the results
Now we can combine the results of the three integrals:
\[ I = \frac{2}{3} - 14 + 10 \]
Calculating this gives:
\[ I = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{-\frac{10}{3}} \]