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Section 3.5 Fonctions trigonométriques

Calculez les intégrales suivantes.

  1. \(\displaystyle \displaystyle\int\cos^3x\;dx\)
  2. \(\displaystyle \displaystyle\int\sin^5x\cos^2x\;dx\)
  3. \(\displaystyle \displaystyle\int\sin^4x\;dx\)
  4. \(\displaystyle \displaystyle\int\tan^6x\sec^4x\;dx\)
  5. \(\displaystyle \displaystyle\int\tan^5x\sec^7x\;dx\)
Réponse
  1. \(\displaystyle \displaystyle\sin x-\frac{\sin^3x}{3}+C\)
  2. \(\displaystyle \displaystyle-\frac{1}{3}\cos^3x+\frac{2}{5}\cos^5x-\frac{1}{7}\cos^7x+C\)
  3. \(\displaystyle \displaystyle\frac{1}{4}\left(\frac{3}{2}x-\sin(2x)+\frac{1}{8}\sin(4x)\right)+C\)
  4. \(\displaystyle \displaystyle\frac{1}{7}\tan^7x+\frac{1}{9}\tan^9x+C\)
  5. \(\displaystyle \displaystyle\frac{1}{11}\sec^{11}x-\frac{2}{9}\sec^9x+\frac{1}{7}\sec^7x+C\)

Déterminez l'intégrale indéfinie.

  1. \(\displaystyle \displaystyle\int\sin^3x\;dx\)
  2. \(\displaystyle \displaystyle\int\cos^4x\;dx\)
  3. \(\displaystyle \displaystyle\int\tan^4x\;dx\)
  4. \(\displaystyle \displaystyle\int\csc x\;dx\)
Réponse
  1. \(\displaystyle \displaystyle\frac{\cos^3x}{3}-\cos x+C\)
  2. \(\displaystyle \displaystyle\frac{\sin(4x)+8\sin(2x)+12x}{32}+C\)
  3. \(\displaystyle \displaystyle\frac{\tan^3x}{3}-\tan x+x+C\)
  4. \(\displaystyle \displaystyle\ln\left|\tan\left(\frac{x}{2}\right)\right|+C\)