Cvičenia

Použite naznačenie metódy per partes na výpočet integrálov.
63. $\int \ln x\,dx, \qquad u' = 1, \quad v = \ln x$.
64. $\int \frac{\ln x\,dx}{x^2}, \qquad u' = \frac{1}{x^2}, \qquad v = \ln x$.
65. $\int x \cos x\,dx, \qquad u' = \cos x, \qquad v = x$.
66. $\int x e^{-2x}\,dx, \qquad u' = e^{-2x}, \qquad v =x$.
67. $\int \mbox{arccotg}\,x\,dx, \qquad u' = 1, \qquad v = \mbox{arccotg}\,x$.
68. $\int \frac{x}{\sin^2 x}\,dx, \qquad u'= \frac{1}{\sin^2 x}, \qquad v = x$.
69. $\int \frac{x \cos x}{\sin^3 x}\,dx, \qquad u' = \frac{\cos x}{\sin^3 x}, \qquad v = x$.
70. $\int x \sinh x\,dx, \qquad u' = \sinh x, \qquad v = x$.
71. $\int \sqrt{1-x^2}\,dx, \qquad u' = 1, \qquad v = \sqrt{1-x^2}$.
72. $\int x \mbox{tg}\,^2 x\,dx, \qquad u' = \mbox{tg}\,^2 x, \qquad v = x$.

Použitím metódy per partes vypočítajte integrály.

 73. $\int x \ln x\,dx$. 		 74. $\int x \sin 3x\,dx$. 


 75. $\int 5x e^{-4x}\,dx$. 		 76. $\int x\ \mbox{arctg}\,x\,dx$. 


 77. $\int \arccos x\,dx$. 		 78. $\int x \cosh x\,dx$. 


 79. $\int (2x+1) \cos(\frac{\pi}{3}-5x)\,dx$. 		 80. $\int \frac{x\,dx}{5^x}$. 


 81. $\int \frac{\ln x}{\sqrt{x}}\,dx$. 		 82. $\int 4x^3 \ln (x^5)\,dx$. 

Opakovaným použitím metódy per partes vypočítajte integrály.

 83. $\int x^2 \sin x\,dx$. 		 84. $\int e^x \cos 2x\,dx$. 


 85. $\int (x^2 + 5) \cos x\,dx$. 		 86. $\int x^2 \sinh x\,dx$. 


 87. $\int (x^2 - 2x + 5) e^{-x}\,dx$. 		 88. $\int x \ln^2 x\,dx$. 


 89. $\int \ln^2 x\,dx$. 		 90. $\int e^{-2x}\sin \frac{x}{2}\,dx$. 


 91. $\int \sin(\ln x)\,dx$. 		 92. $\int x^2 e^{3x}\,dx$. 


 93. $\int (x^2 + 5x + 6) \cos 2x\,dx$. 		 94. $\int x^3 \cos x\,dx$.