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Indefinite integrals of even powers of sines, cosines and tangents

Indefinite integrals of even powers of sines, cosines and tangents

IB Year 6 | Grade 12

Find

  1. \(\displaystyle \displaystyle \displaystyle\int \cos^{2} 3x \, \mathrm{d}x \);
  2. \(\displaystyle \displaystyle \displaystyle\int \tan^{2} \frac{x}{2} \, \mathrm{d}x\);
  3. \( \displaystyle \displaystyle \displaystyle\int \sin^4 x \, \mathrm{d}x\).

Solution:

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Answer:(a) \(\frac{x}{2}+\frac{\sin6x}{12}+C\). (b) \(2\tan\frac{x}{2}-x+C\), on intervals avoiding odd multiples of \(\pi\). (c) \(\frac{3x}{8}-\frac{\sin2x}{4}+\frac{\sin4x}{32}+C\).

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