Evaluate the following integration
`int cos^(3)x dx`

Answer

To evaluate the integral \( \int \cos^3 x \, dx \), we can follow these steps:

### Step 1: Use the trigonometric identity
We know from trigonometric identities that:
\[ \cos 3x = 4 \cos^3 x - 3 \cos x \]
From this, we can express \( \cos^3 x \) in terms of \( \cos 3x \):
\[ \cos^3 x = \frac{1}{4} (\cos 3x + 3 \cos x) \]

### Step 2: Substitute into the integral
Now, we can substitute this expression into our integral:
\[ I = \int \cos^3 x \, dx = \int \frac{1}{4} (\cos 3x + 3 \cos x) \, dx \]
This simplifies to:
\[ I = \frac{1}{4} \int (3 \cos x + \cos 3x) \, dx \]

### Step 3: Break down the integral
We can break this integral into two separate integrals:
\[ I = \frac{1}{4} \left( 3 \int \cos x \, dx + \int \cos 3x \, dx \right) \]

### Step 4: Evaluate the integrals
Now we can evaluate each integral:
1. The integral of \( \cos x \):
\[ \int \cos x \, dx = \sin x \]
2. The integral of \( \cos 3x \):
\[ \int \cos 3x \, dx = \frac{1}{3} \sin 3x \]

### Step 5: Combine the results
Substituting these results back into our expression for \( I \):
\[ I = \frac{1}{4} \left( 3 \sin x + \frac{1}{3} \sin 3x \right) \]
This simplifies to:
\[ I = \frac{3}{4} \sin x + \frac{1}{12} \sin 3x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result for the integral \( \int \cos^3 x \, dx \) is:
\[ \int \cos^3 x \, dx = \frac{3}{4} \sin x + \frac{1}{12} \sin 3x + C \]