Cosine integrals raised to the fourth power
For an integral like this, where you have a trigonometric function raised to a power, a common approach is to use power-reduction identities. These identities help express higher powers of cosine in terms of lower powers, making the integration process easier.
For cosine raised to an even power, you can use the identity that rewrites cosine squared in terms of a double angle. In this case, because you have cosine to the fourth power, you would apply this identity twice. First, break down cosine to the fourth power into a product of two cosine squared terms. Then, use the identity for cosine squared to simplify each term.
Once the expression is simplified, the remaining integral will involve a simpler trigonometric function, which can be integrated directly. This process of reducing the powers using identities is a key technique when handling integrals with higher powers of sine or cosine.