Integral of cosine cubed
For this integral, you're dealing with the cosine function raised to the third power, which introduces a bit more complexity. The key to solving problems like this is to break the function down into something more manageable using trigonometric identities.
One common technique involves splitting the cosine cubed into a product of cosine and cosine squared. Once you've done that, the cosine squared can be rewritten using a power-reduction identity, which expresses it in terms of a simpler trigonometric function. By applying this identity, the original integral simplifies into one part that can be integrated directly and another part that requires substitution.
After splitting up the terms and integrating, the result combines the effects of trigonometric manipulation and substitution. This process is typical when working with integrals of odd powers of sine or cosine functions, where breaking down the function helps reduce it to a more solvable form.