Evaluate `int x cos x dx.`

Answer

To evaluate the integral \( \int x \cos x \, dx \), 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 \( dv \)

According to the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), we choose:

- \( u = x \) (Algebraic)
- \( dv = \cos x \, dx \) (Trigonometric)

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

Now we differentiate \( u \) and integrate \( dv \):

- \( du = dx \)
- \( v = \int \cos x \, dx = \sin x \)

### Step 3: Apply the integration by parts formula

Now we substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:

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

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

Now we need to evaluate \( \int \sin x \, dx \):

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

### Step 5: Substitute back into the equation

Substituting this back into our equation gives:

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

This simplifies to:

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

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

### Final Answer

Thus, the evaluated integral is:

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

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