Techniques of Integration - Integrating Powers of Sine and Cosine

IB AA SL: core and optional content

Core: part (b) and the cos²(4x) term of part (a), using the cosine double-angle formula. Optional: the tan²(5x) term of part (a) and all of part (c), which need the integral of sec².

Integrate the following expressions with respect to \(x\):

1. \(\cos^2 4x + \tan^2 5x\)
2. \(\cos^2 5x - \sin^2 5x\)
3. \(\frac{\sin^2 3x + 5}{\cos^2 3x}\)