Cvičenia

Použitím algebraickej úpravy (ak je potrebná) a substitúcie lineárnej funkcie vypočítajte integrály.

 19. $\int \sin 3x\,dx$. 		 20. $\int \frac{dx}{5-3x}$. 


 21. $\int e^{3-2x}\,dx$. 		 22. $\int \sqrt[3]{3x-2}\,dx$. 


 23. $\int (4 - 7x)^{11}\,dx$. 		 24. $\int \frac{dx}{\cos^2 5x}$. 


 25. $\int \frac{dx}{\sqrt{9-x^2}}$. 		 26. $\int \frac{dx}{x^2+16}$. 

Použitím naznačenej substitúcie vypočítajte integrály.
27. $\int \frac{x\,dx}{\sqrt{x^2 - 4}}, \qquad t = x^2 - 4$.
28. $\int \frac{\cos x}{1 + \sin x}\,dx, \qquad t = \sin x$.
29. $\int \sqrt{\cos^3 x} \sin x\,dx, \qquad t = \cos x$.
30. $\int x e^{x^2}\,dx, \qquad t = x^2$.
31. $\int \frac{dx}{x \ln x}, \qquad t = \ln x$.
32. $\int x^2 \sqrt{x^3 + 1}\,dx, \qquad t = x^3 + 1$.
33. $\int \frac{dx}{\sqrt{x}(x+ 4)}, \qquad t = \frac{\sqrt{x}}{2}$.
34. $\int \frac{x\,dx}{1 + x^4}, \qquad t = x^2$.
35. $\int \frac{dx}{e^x - 1}, \qquad t = e^{-x}$.
36. $\int \frac{e^x \sqrt{\mbox{arctg}\,e^x}}{1 + e^{2x}}\,dx, \qquad t = \mbox{arctg}\,e^x$.
37. $\int \frac{dx}{x \sqrt{x^2-1}}, \qquad t = \frac{1}{x}$.
38. $\int \frac{x\,dx}{\sqrt{x+1}}, \qquad t = \sqrt{x+1}$.

Použitím substitučnej metódy vypočítajte integrály.

 39. $\int \sqrt{4x - 11}\,dx$. 		 40. $\int \frac{6\,dx}{5 - 3x}$. 


 41. $\int \frac{4x}{4 + x^2}\,dx$. 		 42. $\int \frac{14\,dx}{(2x + 3)^8}$. 


 43. $\int 10 x (x^2 + 7)^4\,dx$. 		 44. $\int \frac{x\,dx}{\sqrt{3-x^2}}$. 


 45. $\int \frac{x^2}{1 + x^6}\,dx$. 		 46. $\int x \sqrt[5]{4-x^2}\,dx$. 


 47. $\int \sin^6 x \cos x\,dx$. 		 48. $\int \frac{\sin x}{\sqrt{2 + \cos x}}\,dx$. 


 49. $\int \frac{dx}{x^2+2x+2}$. 		 50. $\int \frac{dx}{\sqrt{4x - 4x^2}}$. 


 51. $\int \frac{e^{\frac{1}{x}}}{x^2}\,dx$. 		 52. $\int (x+2) e^{x^2 + 4x - 5}\,dx$. 


 53. $\int \frac{\ln^4 x}{x}\,dx$. 		 54. $\int \frac{\cos(\ln x)}{x}\,dx$. 


 55. $\int e^{\cos^2 x} \sin 2x\,dx$. 		 56. $\int \frac{\mbox{cotg}\,\sqrt{x}}{\sqrt{x}}\,dx$. 


 57. $\int \frac{\sqrt[3]{\mbox{tg}\,^2 x}}{\cos^2 x}\,dx$. 		 58. $\int \frac{dx}{\sin^2 x \sqrt{\mbox{cotg}\,x - 1}}$. 


 59. $\int \frac{2^x}{\sqrt{1 - 4^x}}\,dx$. 		 60. $\int \frac{e^{2x}}{4+e^x}\,dx$. 


 61. $\int \frac{dx}{(1+x^2)\mbox{arctg}\,x}$. 		 62. $\int \frac{3\,dx}{x \sqrt{1 - \ln^2 x}}$.