`int_(1)^(5) (x^(2) -2x) dx`

Answer

To solve the integral \( \int_{1}^{5} (x^{2} - 2x) \, dx \), we will follow these steps:

### Step 1: Write the integral
We start by writing down the integral we need to solve:
\[ I = \int_{1}^{5} (x^{2} - 2x) \, dx \]

### Step 2: Split the integral
We can split the integral into two separate integrals:
\[ I = \int_{1}^{5} x^{2} \, dx - \int_{1}^{5} 2x \, dx \]

### Step 3: Integrate \(x^{2}\)
Using the power rule for integration, we find:
\[ \int x^{2} \, dx = \frac{x^{3}}{3} \]
So,
\[ \int_{1}^{5} x^{2} \, dx = \left[ \frac{x^{3}}{3} \right]_{1}^{5} = \frac{5^{3}}{3} - \frac{1^{3}}{3} = \frac{125}{3} - \frac{1}{3} = \frac{124}{3} \]

### Step 4: Integrate \(2x\)
Now we integrate \(2x\):
\[ \int 2x \, dx = x^{2} \]
Thus,
\[ \int_{1}^{5} 2x \, dx = \left[ x^{2} \right]_{1}^{5} = 5^{2} - 1^{2} = 25 - 1 = 24 \]

### Step 5: Combine the results
Now we combine the results from the two integrals:
\[ I = \frac{124}{3} - 24 \]
To subtract, we convert \(24\) into a fraction with a denominator of \(3\):
\[ 24 = \frac{72}{3} \]
So,
\[ I = \frac{124}{3} - \frac{72}{3} = \frac{124 - 72}{3} = \frac{52}{3} \]

### Final Answer
The value of the integral is:
\[ I = \frac{52}{3} \]

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