Nature des intégrales généralisées

Étudier la nature (convergence/divergence) des intĂ©grales suivantes :

  1. \(\displaystyle \int_{1}^{+\infty} \ln\left(\frac{x^2 + 1}{x^2 - 1}\right) dx\)
  2. \(\displaystyle \int_{0}^{+\infty} \left(\sqrt[3]{x^3 + 1} - x\right) dx\)
  3. \(\displaystyle \int_{0}^{\frac{\pi}{2}} \sqrt{\tan x} \, dx\)
  4. \(\displaystyle \int_{0}^{+\infty} \frac{x}{e^x - 1} \, dx\)
  5. \(\displaystyle \int_{1}^{+\infty} \frac{dt}{\sqrt[3]{t^5 - 1}}\)
  6. \(\displaystyle \int_{0}^{\frac{\pi}{2}} \ln(\tan x) \, dx\)
  7. \(\displaystyle \int_{1}^{+\infty} \left(\sqrt{x^2 + 2x + 2} - x - 1\right) dx\)
  8. \(\displaystyle \int_{0}^{1} \frac{dx}{1 - \sqrt{x}}\)
  9. \(\displaystyle \int_{0}^{1} \frac{1 - \cos x}{x^{\frac{5}{2}}} \, dx\)
  10. \(\displaystyle \int_{0}^{+\infty} \frac{(\ln x)^2 e^{-x}}{\sqrt{x}} \, dx\)
  11. \(\displaystyle \int_{0}^{+\infty} \frac{dt}{e^t + t^2 e^{-t}}\)
  12. \(\displaystyle \int_{0}^{+\infty} \frac{\ln(1 + \sqrt{x})}{x\sqrt{1 + x^2}} \, dx\)
  13. \(\displaystyle \int_{0}^{+\infty} \frac{dx}{\sqrt{x(1 + e^x)}}\)
  14. \(\displaystyle \int_{0}^{+\infty} \frac{x^2}{\sinh x} \, dx\)
  15. \(\displaystyle \int_{0}^{+\infty} \frac{\sqrt[3]{x + 1} - \sqrt[3]{x}}{\sqrt{x}} \, dx\)
  16. \(\displaystyle \int_{1}^{+\infty} \frac{\ln\left(1 + \frac{1}{x}\right)}{\sqrt{x^2 - 1}} \, dx\)
  17. \(\displaystyle \int_{0}^{1} \frac{(\ln x)^2}{x^{\frac{2}{3}}(1 - x)^{\frac{5}{2}}} \, dx\)
  18. \(\displaystyle \int_{0}^{+\infty} e^{-(\ln t)^2} \, dt\)
  19. \(\displaystyle \int_{0}^{+\infty} \frac{\arctan x}{x(1 + x^2)} \, dx\)
  20. \(\displaystyle \int_{0}^{+\infty} e^{-x}\left(\frac{1}{1 - e^{-x}} - \frac{1}{x}\right) \, dx\)
  21. \(\displaystyle \int_{0}^{+\infty} \frac{x - \arctan x}{x(1 + x^2)\arctan x} \, dx\)
  22. \(\displaystyle \int_{\frac{\pi}{2}}^{+\infty} \ln\left(\cos\left(\frac{1}{x}\right)\right) \, dx\)
  23. \(\displaystyle \int_{0}^{1} \ln(t - t^2) \, dt\)
  24. \(\displaystyle \int_{0}^{+\infty} \ln(t)e^{-t} \, dt\)
  25. \(\displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1 - x^6}}\)
  26. \(\displaystyle \int_{-\infty}^{+\infty} \frac{1 + x^2 e^{-x}}{x^2 + e^{-2x}} \, dx\)
  27. \(\displaystyle \int_{0}^{1} \frac{\ln x}{x^3 + x^2} \, dx\)