Question

Evaluate the integral \[ \int \cos \left( \frac{\pi x}{6} \right) \cos \left( \frac{\pi x}{3} \right) \, dx. \]

• A. \( \frac{1}{6} \sin \left( \frac{\pi x}{3} \right) + c \)
• B. \( \frac{1}{3} \sin \left( \frac{\pi x}{3} \right) + c \)
• C. \( \frac{1}{2} \sin \left( \frac{\pi x}{2} \right) + c \)
• D. \( \frac{1}{4} \sin \left( \frac{\pi x}{2} \right) + c \)

Hint:

To simplify integrals involving products of trigonometric functions, use the product-to-sum identities and then integrate term by term.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks for the integral of a product of trigonometric functions. We can simplify this using the double angle formula for sine.
Step 2: Key Formula or Approach:
Use the identity \( 2 \sin A \cos A = \sin 2A \).
Step 3: Detailed Explanation:
Let the integral be \( I \):
\[ I = \int \sin \left( \frac{x}{16} \right) \cos \left( \frac{x}{16} \right) \cos \left( \frac{x}{8} \right) \cos \left( \frac{x}{4} \right) dx \]
Multiply and divide by 2:
\[ I = \int \frac{1}{2} \left[ 2 \sin \left( \frac{x}{16} \right) \cos \left( \frac{x}{16} \right) \right] \cos \left( \frac{x}{8} \right) \cos \left( \frac{x}{4} \right) dx \]
\[ I = \int \frac{1}{2} \sin \left( \frac{x}{8} \right) \cos \left( \frac{x}{8} \right) \cos \left( \frac{x}{4} \right) dx \]
Repeat the process by multiplying and dividing by 2 again:
\[ I = \int \frac{1}{4} \left[ 2 \sin \left( \frac{x}{8} \right) \cos \left( \frac{x}{8} \right) \right] \cos \left( \frac{x}{4} \right) dx \]
\[ I = \int \frac{1}{4} \sin \left( \frac{x}{4} \right) \cos \left( \frac{x}{4} \right) dx \]
One more time:
\[ I = \int \frac{1}{8} \left[ 2 \sin \left( \frac{x}{4} \right) \cos \left( \frac{x}{4} \right) \right] dx = \int \frac{1}{8} \sin \left( \frac{x}{2} \right) dx \]
Now, integrate:
\[ I = \frac{1}{8} \left[ \frac{-\cos(x/2)}{1/2} \right] + c \]
\[ I = \frac{1}{8} \cdot (-2) \cos \left( \frac{x}{2} \right) + c = -\frac{1}{4} \cos \left( \frac{x}{2} \right) + c \]
Step 4: Final Answer:
The integral is \( -\frac{\cos(x/2)}{4} + c \).