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Section 3.2 Formules d'intégration de base

Calculez les intégrales indéfinies suivantes.

  1. \(\displaystyle\int\left(x^3+\pi+x^{-3}\right)\;dx\)

  2. \(\displaystyle\int(t+3)(2t+5)\;dt\)

  3. \(\displaystyle\int\left(e+x+\sqrt{x}-4\sqrt[3]{x^5}\right)\;dx\)

  4. \(\displaystyle\int\frac{x^4-3\sqrt{x}}{x}\;dx\)

  5. \(\displaystyle\int\left(\sec s+\tan s\right)\sec s\;ds\)

  6. \(\displaystyle\int\frac{\sin 2x}{\sin x}\;dx\)

  7. \(\displaystyle\int\frac{\sin^2 x}{\sin x}\;dx\)

  8. \(\displaystyle\int(x^3+1)x^2\;dx\)

  9. \(\displaystyle\int\frac{-11}{\sin^2x}\;dx\)

  10. \(\displaystyle\int(x^3-x^2+1)\sqrt{x}\;dx\)

Réponse

  1. \(\displaystyle\frac{x^4}{4}+\pi x-\frac{1}{2x^2}+C\)

  2. \(\displaystyle\frac{2t^3}{3}+\frac{11t^2}{2}+15t+C\)

  3. \(\displaystyle ex+\frac{x^2}{2}+\frac{2x^\frac{3}{2}}{3}-\frac{3x^\frac{8}{3}}{2}+C\)

  4. \(\displaystyle\frac{1}{4}x^4-6\sqrt{x}+C\)

  5. \(\displaystyle\sec s+\tan s+C\)

  6. \(\displaystyle 2\sin x+C\)

  7. \(\displaystyle-\cos x+C\)

  8. \(\displaystyle\frac{x^6}{6}+\frac{x^3}{3}+C\)

  9. \(\displaystyle 11\cot x+C\)

  10. \(\displaystyle\frac{2x^\frac{9}{2}}{9}-\frac{2x^\frac{7}{2}}{7}+\frac{2x^\frac{3}{2}}{3}+C\)

Calculez les intégrales indéfinies suivantes.

  1. \(\displaystyle\int\frac{(x^{-2}+x)^2}{x}\;dx\)

  2. \(\displaystyle\int\frac{2x^3+x^2+2x+4}{5x^2+5}\;dx\)

  3. \(\displaystyle\int 7^{5z}\;dz\)

  4. \(\displaystyle\int(e^x+e^{-x})^2\;dx\)

  5. \(\displaystyle\int\frac{3}{\sin t}\;dt\)

  6. \(\displaystyle\int\frac{6}{\mathrm{cotan}\theta}\;d\theta\)

  7. \(\displaystyle\int\frac{1}{\cos^2y}\;dy\)

  8. \(\displaystyle\int\frac{2}{1-\cos^2t}\;dt\)

  9. \(\displaystyle\int\frac{1}{1-\cos\theta}\;d\theta\)

  10. \(\displaystyle\int\tan^2x\;dx\)

Réponse

  1. \(\displaystyle-\frac{1}{4x^4}-\frac{2}{x}+\frac{x^2}{2}+C\)

  2. \(\displaystyle\frac{x^2}{5}+\frac{x}{5}+\frac{3}{5}\arctan(x)+C\)

  3. \(\displaystyle\frac{7^{5z}}{5\ln(7)}+C\)

  4. \(\displaystyle\frac{e^{2x}}{2}+2x-\frac{1}{2e^{2x}}+C\)

  5. \(\displaystyle-3\ln|\mathrm{cosec}(t)+\mathrm{cotan}(t)|+C\)

  6. \(\displaystyle 6\ln|\sec\theta|+C\)

  7. \(\displaystyle\tan y+C\)

  8. \(\displaystyle-2\mathrm{cotan}t+C\)

  9. \(\displaystyle-\mathrm{cotan}\theta-\mathrm{cosec}\theta+C\)

  10. \(\displaystyle\tan x-x+C\)

Calculez les intégrales définies suivantes.

  1. \(\displaystyle\int_{-1}^4|3x-4|\;dx\)

  2. \(\displaystyle\int_0^{\pi/4}\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)\;dx\)

Réponse

  1. \(\displaystyle\frac{113}{6}\)

  2. \(\displaystyle\frac{1}{2}-\frac{\sqrt{2}}{4}\)