Techniques of Integration - Integrating Powers of Sine and Cosine

IB AA SL: core and optional content

Core: parts (a) and (b), and the √(1−2x) term of part (c). Optional: the tan²(1−2x) term of part (c), which needs the integral of sec².

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

1. \(\displaystyle\displaystyle \displaystyle\int \sin^2 4x + \sqrt{5x} \, \mathrm{d}x\)
2. \(\displaystyle\displaystyle \displaystyle\int \cos^2 (3x + 1) - \cos\ (3x+1)\,\mathrm{d}x\)
3. \(\displaystyle\displaystyle \displaystyle\int \tan^2 (1 - 2x) + \sqrt{1-2x}\,\mathrm{d}x\)