Find the value of ∫ Cos x + x dx.

Explanation

To solve the integral \( \int (\cos x + x) \, dx \), we can break it down into two separate integrals. Here’s the step-by-step solution:

### Step 1: Split the Integral
We start by splitting the integral into two parts:
\[ \int (\cos x + x) \, dx = \int \cos x \, dx + \int x \, dx \]

### Step 2: Integrate \( \cos x \)
We know that the integral of \( \cos x \) is:
\[ \int \cos x \, dx = \sin x + C_1 \]
where \( C_1 \) is a constant of integration.

### Step 3: Integrate \( x \)
Next, we use the power rule for integration to find the integral of \( x \):
\[ \int x \, dx = \frac{x^2}{2} + C_2 \]
where \( C_2 \) is another constant of integration.

### Step 4: Combine the Results
Now we combine the results from Step 2 and Step 3:
\[ \int (\cos x + x) \, dx = \sin x + C_1 + \frac{x^2}{2} + C_2 \]
Since \( C_1 \) and \( C_2 \) are both constants, we can combine them into a single constant \( C \):
\[ \int (\cos x + x) \, dx = \sin x + \frac{x^2}{2} + C \]

### Final Answer
Thus, the final answer is:
\[ \int (\cos x + x) \, dx = \sin x + \frac{x^2}{2} + C \]

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