The primitive of the function `f(x)= x | cos x|`, when `pi/2 lt x lt pi` is given by

Options

A. `cos x + x sin x +C`
B. `-cos x - x sin x +C`
C. `x sin x - cos x +C`
D. None of the above

Answer

To find the primitive (indefinite integral) of the function \( f(x) = x \cos x \) for \( \frac{\pi}{2} < x < \pi \), we will follow these steps:

### Step 1: Set up the integral
We need to integrate the function:
\[ \int x \cos x \, dx \]

### Step 2: Use integration by parts
We will apply the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Let:
- \( u = x \) (thus \( du = dx \))
- \( dv = \cos x \, dx \) (thus \( v = \sin x \))

### Step 3: Apply the integration by parts
Now, substituting into the integration by parts formula:
\[ \int x \cos x \, dx = x \sin x - \int \sin x \, dx \]

### Step 4: Integrate \( \sin x \)
The integral of \( \sin x \) is:
\[ \int \sin x \, dx = -\cos x \]
So, substituting this back into our equation:
\[ \int x \cos x \, dx = x \sin x + \cos x + C \]

### Step 5: Final expression
Thus, the primitive of the function \( f(x) = x \cos x \) for \( \frac{\pi}{2} < x < \pi \) is:
\[ \int x \cos x \, dx = x \sin x + \cos x + C \]