`intcos^(3)x" " dx=`

Options

A. `(-3sinx)/(4)+(sin3x)/(12)+c`
B. `(3sinx)/(4)-(sin3x)/(12)+c`
C. `(-4sinx)/(3)+(sin3x)/(12)+c`
D. `(4sinx)/(3)-(sin3x)/(12)+c`

Answer

To solve the integral \( \int \cos^3 x \, dx \), we can use a trigonometric identity to simplify the expression. Here’s a step-by-step solution:

### Step 1: Use the Trigonometric Identity
We know from trigonometric identities that:
\[ \cos 3x = 4 \cos^3 x - 3 \cos x \]
Rearranging this gives us:
\[ \cos^3 x = \frac{1}{4} (\cos 3x + 3 \cos x) \]

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

### Step 3: Split the Integral
We can split the integral into two parts:
\[ \frac{1}{4} \left( \int \cos 3x \, dx + 3 \int \cos x \, dx \right) \]

### Step 4: Integrate Each Part
Now we integrate each part separately:
1. For \( \int \cos 3x \, dx \):
\[ \int \cos 3x \, dx = \frac{1}{3} \sin 3x \]
2. For \( \int \cos x \, dx \):
\[ \int \cos x \, dx = \sin x \]

### Step 5: Combine the Results
Now substituting back into our expression:
\[ \frac{1}{4} \left( \frac{1}{3} \sin 3x + 3 \sin x \right) + C \]
This simplifies to:
\[ \frac{1}{12} \sin 3x + \frac{3}{4} \sin x + C \]

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