`int_2^5 (x^2+x)dx`

Answer

To solve the integral \( \int_2^5 (x^2 + x) \, dx \), we can break it down into two separate integrals:

\[ \int_2^5 (x^2 + x) \, dx = \int_2^5 x^2 \, dx + \int_2^5 x \, dx \]

### Step 1: Integrate \( x^2 \)
The integral of \( x^2 \) is calculated as follows:

\[ \int x^2 \, dx = \frac{x^3}{3} + C \]

Now, we will evaluate this from 2 to 5:

\[ \int_2^5 x^2 \, dx = \left[ \frac{x^3}{3} \right]_2^5 = \frac{5^3}{3} - \frac{2^3}{3} \]

Calculating the values:

\[ = \frac{125}{3} - \frac{8}{3} = \frac{125 - 8}{3} = \frac{117}{3} \]

### Step 2: Integrate \( x \)
Next, we integrate \( x \):

\[ \int x \, dx = \frac{x^2}{2} + C \]

Now, we will evaluate this from 2 to 5:

\[ \int_2^5 x \, dx = \left[ \frac{x^2}{2} \right]_2^5 = \frac{5^2}{2} - \frac{2^2}{2} \]

Calculating the values:

\[ = \frac{25}{2} - \frac{4}{2} = \frac{25 - 4}{2} = \frac{21}{2} \]

### Step 3: Combine the Results
Now we will combine the results of both integrals:

\[ \int_2^5 (x^2 + x) \, dx = \frac{117}{3} + \frac{21}{2} \]

To add these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6.

Rewriting the fractions:

\[ \frac{117}{3} = \frac{117 \times 2}{3 \times 2} = \frac{234}{6} \]
\[ \frac{21}{2} = \frac{21 \times 3}{2 \times 3} = \frac{63}{6} \]

Now, adding them together:

\[ \frac{234}{6} + \frac{63}{6} = \frac{234 + 63}{6} = \frac{297}{6} \]

### Step 4: Simplify the Result
Finally, we can simplify \( \frac{297}{6} \):

\[ \frac{297}{6} = 49.5 \]

Thus, the final answer is:

\[ \int_2^5 (x^2 + x) \, dx = 49.5 \]