Solve the following integration: ∫ cos 2x cos 4x cos 6x dx.

To solve the integral ∫ cos(2x) cos(4x) cos(6x) dx, we can use a product-to-sum identity to simplify the expression. The product-to-sum identities help convert products of cosines into sums, making integration easier.

Step 1: Apply Product-to-Sum Identities

We can start by combining two of the cosine functions. Let's first combine cos(2x) and cos(6x):

• cos(A) cos(B) = 1/2 [cos(A + B) + cos(A - B)

Using this identity:

cos(2x) cos(6x) = 1/2 [cos(8x) + cos(4x)]

Step 2: Substitute Back into the Integral

Now, we can rewrite the integral:

∫ cos(2x) cos(4x) cos(6x) dx = ∫ (1/2 [cos(8x) + cos(4x)]) cos(4x) dx

Step 3: Expand the Integral

Next, we can distribute cos(4x):

∫ (1/2 [cos(8x) cos(4x) + cos(4x) cos(4x)]) dx

Now, apply the product-to-sum identity again for cos(8x) cos(4x):

cos(8x) cos(4x) = 1/2 [cos(12x) + cos(4x)]

Step 4: Combine and Simplify

Substituting this back, we have:

∫ (1/2 [1/2 (cos(12x) + cos(4x)) + cos^2(4x)]) dx

Now, we can simplify further:

∫ (1/4 cos(12x) + 1/4 cos(4x) + 1/2 (1 + cos(8x))) dx

Step 5: Integrate Each Term

Now, we can integrate each term separately:

• ∫ cos(12x) dx = (1/12) sin(12x)
• ∫ cos(4x) dx = (1/4) sin(4x)
• ∫ cos(8x) dx = (1/8) sin(8x)

Putting it all together, we have:

∫ cos(2x) cos(4x) cos(6x) dx = (1/48) sin(12x) + (1/16) sin(4x) + (1/16) x + C

Final Result

The final answer for the integral is:

∫ cos(2x) cos(4x) cos(6x) dx = (1/48) sin(12x) + (1/16) sin(4x) + (1/16) x + C