integrate the function `x sin x`

Answer

To integrate the function \( x \sin x \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)

We choose:
- \( u = x \) (which we will differentiate)
- \( dv = \sin x \, dx \) (which we will integrate)

### Step 2: Differentiate \( u \) and Integrate \( dv \)

Now we differentiate \( u \) and integrate \( dv \):
- \( du = dx \)
- \( v = \int \sin x \, dx = -\cos x \)

### Step 3: Apply the Integration by Parts Formula

Now we apply the integration by parts formula:

\[ \int x \sin x \, dx = uv - \int v \, du \]

Substituting the values we found:

\[ = x(-\cos x) - \int (-\cos x) \, dx \]

This simplifies to:

\[ = -x \cos x + \int \cos x \, dx \]

### Step 4: Integrate \( \cos x \)

Now we integrate \( \cos x \):

\[ \int \cos x \, dx = \sin x \]

### Step 5: Combine the Results

Putting it all together, we have:

\[ \int x \sin x \, dx = -x \cos x + \sin x + C \]

where \( C \) is the constant of integration.

### Final Answer

Thus, the integral of \( x \sin x \) is:

\[ \int x \sin x \, dx = -x \cos x + \sin x + C \]