Výsledky

 63. $x \ln x - x + c$. 		 64. $-\frac{\ln x}{x} - \frac{1}{x} + c$. 


 65. $x \sin x + \cos x + c$. 		 66. $-\frac12 x e^{-2x} - \frac14 e^{-2x} + c$. 


 67. $x \mbox{arccotg}\,x + \frac12 \ln(1+x^2) + c$. 		 68. $-x \mbox{cotg}\,x + \ln \vert\sin x\vert + c$. 


 69. $-\frac{x}{2 \sin^2 x} - \frac12 \mbox{cotg}\,x + c$. 		 70. $x \cosh x - \sinh x + c$. 


 71. $\frac12(x \sqrt{1-x^2} + \arcsin x) + c$. 		 72. $x \mbox{tg}\,x + \ln \vert\cos x\vert - \frac{x^2}{2} + c$. 

Vo výsledkoch nasledujúcich cvičení je ešte pred výsledkom uvedená voľba funkcie $u'$ v metóde per partes, ktorou je možné integrál riešiť. Funkciu $v$ si čitateľ doplní.

 73. $u' = x,\ I = \frac12 x^2 \ln x - \frac14 x^2 + c$. 


 74. $u' = \sin 3x,\ I = -\frac13 x \cos 3x + \frac19 \sin 3x + c$. 


 75. $u' = e^{-4x},\I = -\frac54 x e^{-4x} - \frac{5}{16} e^{-4x} + c$. 


 76. $u' = x,\I = \frac{x^2}{2} \mbox{arctg}\,x - \frac{x}{2} + \frac12 \mbox{arctg}\,x + c$. 


 77. $u' = 1,\ I = x \arccos x - \sqrt{1-x^2} + c$. 


 78. $u' = \cosh x,\ I = x \sinh x - \cosh x + c$. 


 79. $u' = \cos(\frac{\pi}{3} - 5x),\I = -\frac{2x+1}{5}\sin(\frac{\pi}{3} - 5x) +\frac{2}{25} \cos(\frac{\pi}{3} - 5x) + c$. 


 80. $u' = 5^{-x},\I = -\frac{x 5^{-x}}{\ln 5} - \frac{5^{-x}}{\ln^2 5} + c$. 


 81. $u' = \frac{1}{\sqrt{x}},\I = 2 \sqrt{x} \ln x - 4 \sqrt{x} + c$. 


 82. $u' = 4x^3,\ I = 5x^4 \ln x - \frac54 x^4 + c$. 

 83. $u' = \sin x,\ I = -x^2 \cos x + 2x \sin x + 2 \cos x + c$. 


 84. $u'$ je jedno, $I = \frac{e^x}{5}(\cos 2x + 2\sin 2x) + c$. 


 85. $u' = \cos x,\ I = (x^2+3) \sin x + 2x \cos x + c$. 


 86. $u' = \sinh x,\ I = (x^2 + 2) \cosh x - 2x \sinh x + c$. 


 87. $u' = e^{-x},\ I = -e^{-x}(x^2 + 5) + c$. 


 88. $u' = x,\ I = \frac12 x^2 (\ln^2 x - \ln x) + \frac14 x^2 + c$. 


 89. $u' = 1,\ I = x \ln^2 x - 2x \ln x + 2x + c$. 


 90. $u'$ je jedno, $I = -\frac{8}{17} e^{-2x}(\sin \frac{x}{2} +\frac14 \cos \frac{x}{2}) + c$. 


 91. $u' = 1,\ I = \frac{x}{2}\left(\sin(\ln x) - \cos(\ln x)\right) + c$. 


 92. $u' = e^{3x},\ I = \frac{e^{3x}}{27}(9x^2 - 6x + 2) + c$. 


 93. $u' = \cos 2x,\ I = \frac{2x^2+10x+11}{4} \sin 2x +\frac{2x+5}{4} \cos 2x + c$. 


 94. $u' = \cos x,\ I = (x^3 - 6x) \sin x + (3x^2 - 6) \cos x + c$.