`int_0^5 (x+1)dx`

Answer

To solve the integral \( \int_0^5 (x + 1) \, dx \), we will break it down into simpler parts and then evaluate it step by step.

### Step 1: Break down the integral
We can separate the integral into two parts:
\[ \int_0^5 (x + 1) \, dx = \int_0^5 x \, dx + \int_0^5 1 \, dx \]

### Step 2: Evaluate the first integral
Now we will evaluate the first integral \( \int_0^5 x \, dx \). The formula for the integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} + C \]
Applying the limits from 0 to 5:
\[ \int_0^5 x \, dx = \left[ \frac{x^2}{2} \right]_0^5 = \frac{5^2}{2} - \frac{0^2}{2} = \frac{25}{2} - 0 = \frac{25}{2} \]

### Step 3: Evaluate the second integral
Next, we evaluate the second integral \( \int_0^5 1 \, dx \). The integral of 1 is simply:
\[ \int 1 \, dx = x + C \]
Applying the limits from 0 to 5:
\[ \int_0^5 1 \, dx = [x]_0^5 = 5 - 0 = 5 \]

### Step 4: Combine the results
Now we combine the results from both integrals:
\[ \int_0^5 (x + 1) \, dx = \int_0^5 x \, dx + \int_0^5 1 \, dx = \frac{25}{2} + 5 \]
To add these, we can convert 5 into a fraction:
\[ 5 = \frac{10}{2} \]
Thus,
\[ \int_0^5 (x + 1) \, dx = \frac{25}{2} + \frac{10}{2} = \frac{35}{2} \]

### Step 5: Final answer
The final answer can also be expressed as a mixed fraction:
\[ \frac{35}{2} = 17 \frac{1}{2} \]

### Summary of the solution:
The value of the integral \( \int_0^5 (x + 1) \, dx \) is \( \frac{35}{2} \) or \( 17 \frac{1}{2} \).

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