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Section 3.7 Fonctions rationnelles

Associez chaque fonction rationnelle de la colonne de gauche à sa décomposition en fractions partielles parmi les choix proposés dans la colonne de droite.

(a) \(\displaystyle\frac{3x^2+4x+2}{x^3+2x^2+x}\)

(b) \(\displaystyle\frac{x^2+3x-5}{x^3+2x^2-x-2}\)

(c) \(\displaystyle\frac{2x^3-8x^2+9x-1}{x^2-4x+3}\)

(d) \(\displaystyle\frac{x^2+4x+1}{(x+1)(x^2+2x+2)}\)

(I) \(\displaystyle 2x-\frac{1}{x-1}+\frac{4}{x-3}\)

(II) \(\displaystyle\frac{7}{2(x+1)}-\frac{7}{3(x+2)}-\frac{1}{6(x-1)}\)

(III) \(\displaystyle-\frac{2}{x+1}+\frac{3x+5}{x^2+2x+2}\)

(IV) \(\displaystyle\frac{2}{x}+\frac{1}{x+1}-\frac{1}{(x+1)^2}\)

Réponse

(a) - (IV)

(b) - (II)

(c) - (I)

(d) - (III)

Caculez les intégrales indéfinies suivantes.

  1. \(\displaystyle \displaystyle\int\frac{x^3+x}{x-1}\,dx\)
  2. \(\displaystyle \displaystyle\int\frac{1}{x^2-25}\,dx\)
  3. \(\displaystyle \displaystyle\int\frac{x^2+2x-1}{2x^3+3x^2-2x}\,dx\)
  4. \(\displaystyle \displaystyle\int\frac{x^4-2x^2+4x+1}{x^3-x^2-x+1}\,dx\)
  5. \(\displaystyle \displaystyle\int\frac{x^3+x-1}{(x-1)(x-3)^2}\,dx\)
  6. \(\displaystyle \displaystyle\int\frac{1-x-2x^2-x^3}{x(x^2+1)^2}\,dx\)
Réponse
  1. \(\displaystyle \displaystyle\frac{x^3}{3}+\frac{x^2}{2}+2x+2\ln|x-1|+C\)
  2. \(\displaystyle \displaystyle\frac{1}{10}\ln\left|\frac{x-5}{x+5}\right|+C\)
  3. \(\displaystyle \displaystyle\frac{1}{2}\ln\left|x\right|+\frac{1}{10}\ln\left|2x-1\right|-\frac{1}{10}\ln\left|x+2\right|+C\)
  4. \(\displaystyle \displaystyle\frac{x^2}{2}+x-\frac{2}{x-1}+\ln\left|\frac{x-1}{x+1}\right|+C\)
  5. \(\displaystyle \displaystyle x-\frac{29}{2(x-3)}+\frac{27}{4}\ln|x-3|+\frac{1}{4}\ln|x-1|+C\)
  6. \(\displaystyle \displaystyle\ln|x|-\frac{1}{2}\ln(x^2+1)-\arctan(x)-\frac{3}{2(x^2+1)}+C\)