Integrating Powers of Trigonometric Functions

When faced with integrals like

$\int \sin^2(3\theta) \, d\theta \quad \text{or} \quad \int \cos^4\Big(\frac{1}{5}\theta\Big) \, d\theta,$

we encounter a challenge: while we can integrate basic functions like $\sin(3\theta)$ or $\cos\Big(\frac{1}{5}\theta\Big)$ directly, we do not have any techniques to deal with the powers (2 or 4 in these cases).

One key observation is that these powers of sine and cosine are periodic functions., which suggests that we can express their powers in terms of trigonometric functions without exponents. In fact, we can achieve this by using the double angle formulas. If we recall the double angle formula for cosine:

$\cos(2x) = \cos^2(x) - \sin^2(x)$

and the Pythagorean identity,

$\cos^2(x) + \sin^2(x) = 1$

We can combine these to eliminate the powers and solve integrals like these examples.

...