Techniques of Integration - Integrating Powers of Sine and Cosine

Techniques of Integration - Integrating Powers of Sine and Cosine

IB AA SL: core and optional content

Core: part (a), the sine of a linear expression. Optional: parts (b) and (c), which rewrite as integrals of sec².

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

  1. \(\pi \sin \left( \frac{2 + \mathrm{e}x}{3} \right)\)

  2. \(\frac{3-\sin^2 2x}{\cos^2 2x}\)
  3. \(\frac{\tan^2(1-x)[3\sin^2(1-x)-1]}{\sin^2(1-x)}\)

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Answer:(a) \(-\frac{3\pi}{\mathrm e}\cos\left(\frac{2+\mathrm e x}{3}\right)+C\). (b) \(\tan2x+x+C\). (c) \(-2\tan(1-x)-3x+C\).

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