Cvičenia

Vypočítajte určité integrály.

 10. $\int\limits_{0}^{1} \frac{dx}{(2x+1)^3}$, 		 11. $\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos x}{\sin^2 x}\,dx$, 

 12. $\int\limits_{9}^{4} \frac{1 - \sqrt{x}}{\sqrt{x}}\,dx$, 		 13. $\int\limits_{-1}^{0} x\sqrt{1-x^2}\,dx$, 

 14. $\int\limits_{0}^{\pi} x \cos (2x - \frac{\pi}{2})\,dx$, 		 15. $\int\limits_{\sqrt{2}}^{2} \frac{dx}{x\sqrt{x^2-1}}$, 

 16. $\int\limits_{0}^{\frac{\sqrt{2}}{2}} \frac{x\,dx}{\sqrt{1-x^4}}$, 		 17. $\int\limits_{\frac12}^{\frac34} \frac{dx}{\sqrt{x}\sqrt{1-x}}$, 

 18. $\int\limits_{1}^{3} \frac{\cos(\ln x)}{x}\,dx$, 		 19. $\int\limits_{0}^{2} \frac{e^x}{1+e^x}\,dx$, 

 20. $\int\limits_{0}^{1} \frac{3x}{1+x^2}\,dx$, 		 21. $\int\limits_{0}^{\sqrt{\ln 2}} x e^{x^2}\,dx$, 

 22. $\int\limits_{1}^{9} \frac{dx}{\sqrt{x}(1+\sqrt{x})}$, 		 23. $\int\limits_{\frac{\pi^2}{9}}^{\frac{\pi^2}{4}} \frac{\cos \sqrt{x}}{\sqrt{x}}\,dx$, 

 24. $\int\limits_{1}^{4} \frac{2^{\sqrt{x}}}{2 \sqrt{x}}\,dx$, 		 25. $\int\limits_{0}^{1} \sqrt{x^5+2x}(5x^4+2)\,dx$, 

 26. $\int\limits_{0}^{\pi} \sin 2x \cos^2 2x\,dx$, 		 27. $\int\limits_{0}^{1} e^{\sqrt{x}}\,dx$, 

 28. $\int\limits_{0}^{2 \pi} \vert\sin x\vert\,dx$, 		 29. $\int\limits_{0}^{\pi} \cos^2 x\,dx$, 

 30. $\int\limits_{0}^{\frac{2 \pi}{\omega}} \cos(\omega x)\,dx$, 		 31. $\int\limits_{0}^{\frac12} \frac{2 dx}{\sqrt{1 - 4x^2}}$, 

 32. $\int\limits_{0}^{\frac{\pi}{4}} \frac{\sin^2 2x}{1 + \cos 2x}\,dx$, 		 33. $\int\limits_{-\pi}^{0} \sin 3x \cos 2x\,dx$, 

 34. $\int\limits_{0}^{\pi} \cos 3x \cos 4x\,dx$, 		 35. $\int\limits_{-\pi}^{\pi} \sin^2 3x\,dx$, 

 36. $\int\limits_{0}^{\frac12} \frac{dx}{1-x^2}$, 		 37. $\int\limits_{0}^{2 \sqrt{2}} \frac{x^3}{x^2+1}\,dx$, 

 38. $\int\limits_{\frac14}^{\frac34} \frac{dx}{x-x^2}$, 		 39. $\int\limits_{-1}^{0} \frac{x\,dx}{x^2-3x+2}$, 

 40. $\int\limits_{0}^{1} \frac{3x^2}{x^2+2x+1}\,dx$, 		 41. $\int\limits_{0}^{\sqrt{3}} \frac{5x^2}{x^2+1}\,dx$, 

 42. $\int\limits_{-1}^{0} \frac{2x}{(x^2+1)(x-1)^2}\,dx$, 		 43. $\int\limits_{0}^{\ln 2} \frac{e^x}{e^{2x}+e^x+2}\,dx$, 

 44. $\int\limits_{0}^{1} \frac{12\,dx}{\sqrt{4-x^2}}$, 		 45. $\int\limits_{0}^{1} \frac{x^3\,dx}{\sqrt{x^2+1}}$, 

 46. $\int\limits_{\frac34}^{\frac45} \frac{dx}{x^2 \sqrt{1-x^2}}$, 		 47. $\int\limits_{0}^{2} \frac{x\,dx}{x^2+4x+3}$, 

 48. $\int\limits_{0}^{\pi} \sqrt{1 - \cos^2 2x}\,dx$, 		 49. $\int\limits_{0}^{2\pi} \sqrt{\frac{1-\cos x}{2}}\,dx$, 

 50. $\int\limits_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \sqrt{\mbox{cotg}\,^2 x + 1}\,dx$, 		 51. $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos^2 x} - 1}\,dx$, 

 52. $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \mbox{tg}\,^2 x\,dx$, 		 53. $\int\limits_{1}^{2} x \ln x\,dx$, 

 54. $\int\limits_{0}^{1} \ln (x+1)\,dx$, 		 55. $\int\limits_{0}^{\frac{\pi}{2}} x^2 \sin 2x\,dx$, 

 56. $\int\limits_{1}^{e} \frac{\ln x}{x}\,dx$, 		 57. $\int\limits_{0}^{4} (x-1) \sqrt{2x+1}\,dx$, 

 58. $\int\limits_{-\frac{\pi}{3}}^{0} \frac{dx}{\cos^3 x}$, 		 59. $\int\limits_{0}^{1} \ln(x^2+1)\,dx$, 

 60. $\int\limits_{0}^{1} x \mbox{arctg}\,x\,dx$, 		 61. $\int\limits_{0}^{1} \arcsin x\,dx$, 

 62. $\int\limits_{0}^{\pi} x^2 \cos x\,dx$, 		 63. $\int\limits_{0}^{\frac{\pi}{2}} e^x \sin x\,dx$, 

 64. $\int\limits_{0}^{\ln 2} x \cosh x\,dx$, 		 65. $\int\limits_{0}^{1} x^3 e^{2x}\,dx$.