អ្នកបាចវិញ្ញាសារនិងរូបមន្ត: October 2014

មេរៀន រូបមន្ត អាំងតេក្រាល

រូបមន្ត និងទំរង់អាំងតេក្រាលមួយចំនួន ទំរង់ស្តង់ដា share សិរីមង្គល

1. $\int adx=ax+c$
2. $\int af(x)dx=a \int f(x)dx+c$
3. $\int udv=uv- \int vdu$ (អាំងតេក្រាលដោយផ្នែក)
4. $\int u^{n}du= \displaystyle \frac{u^{n+1}}{n+1}+c$
5. $\int \dfrac{du}{u}=ln|u|+c$
6. $\int e^{u}du=e^{u}+c$
7. $\int a^{u}du= \int e^{ulna}du= \displaystyle \frac{e^{ulna}}{lna}= \displaystyle \frac{a^{u}}{lna}+c$
8. $\int sinudu=-cosu+c$
9. $\int cosudu=sinu+c$
10. $\int tanudu=ln|secu|=-ln|cosu|+c$
11. $\int cotudu=ln|sinu|+c$
12. $\int secudu=ln|secu+tanu|+c=ln tan \displaystyle \frac{u}{2}+ \frac{ \pi}{4}+c$
13. $\int csc udu=ln|csc u-cotu|=ln|tan \frac{u}{2}|+c$
14. $\int sec^{2}udu=tanu+c$
15. $\int csc^{2}udu=-csc u+c$
16. $\int sec u tanudu=secu+c$
17. $\int csc ucotudu=-csc u+c$

18. $\int \displaystyle \frac{du}{u^{2}+a^{2}}= \frac{1}{a}acrtg \frac{u}{a}+c$

19. $\int \displaystyle \frac{du}{u^{2}-a^{2}}= \frac{1}{2a}ln \displaystyle | \frac{u-a}{u+a}|+c$

20. $\int \displaystyle \frac{du}{a^{2}-u^{2}}= \displaystyle \frac{1}{2a} ln| \displaystyle \frac{a+u}{a-u}|+c= \displaystyle \frac{1}{a}acrtgh \frac{u}{a}+c$

21. $\int \displaystyle \frac{du}{ \sqrt{a^{2}-u^{2}}}=arcsin \frac{u}{a}+c$

22. $\int \displaystyle \frac{du}{ \sqrt{u^{2}+a^{2}}}=ln(u+ \sqrt{u^{2}+a^{2}})+c$

23. $\int \displaystyle \frac{du}{ \sqrt{u^{2}-a^{2}}}=ln|u+ \sqrt{u^{2}-a^{2}}|+c$

24. $\int \displaystyle \frac{du}{u \sqrt{u^{2}-a^{2}}}= \displaystyle \frac{1}{|a|}arcsec \displaystyle | \frac{u}{a}|+c$

25. $\int \displaystyle \frac{du}{u \sqrt{u^{2}+a^{2}}}=- \displaystyle \frac{1}{a}ln| \displaystyle \frac{a+ \sqrt{u^{2}+a^{2}}}{u}|+c$

26. $\int \displaystyle \frac{du}{u \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{1}{a}ln | \displaystyle \frac{a+ \sqrt{a^{2}-u^{2}}}{u}|+c$
ទំរង់ជាប់ au+b

27. $\int \displaystyle \frac{du}{au+b}= \displaystyle \frac{1}{a}ln|au+b|+c$
28. $\int \displaystyle \frac{udu}{au+b}= \displaystyle \frac{u}{a}- \displaystyle \frac{b}{a^{2}}ln|au+b|+c$
29. $\int \displaystyle \frac{u^{2}du}{au+b}= \displaystyle \frac{(au+b)^{2}}{2a^{3}}- \displaystyle \frac{2b(au+b)}{a^{3}}+ \displaystyle \frac{b^{2}}{a^{3}}ln|au+b|+c$
30. $\int \displaystyle \frac{du}{u(au+b)}= \displaystyle \frac{1}{b}ln| \displaystyle \frac{u}{au+b}|+c$
31. $\int \displaystyle \frac{du}{u^{2}(au+b)}=- \displaystyle \frac{1}{bu}+ \displaystyle \frac{a}{b^{2}}ln| \displaystyle \frac{au+b}{u}|+c$
32. $\int \displaystyle \frac{du}{(au+b)^{2}}= \displaystyle \frac{-1}{a(au+b)}+c$
33. $\int \displaystyle \frac{udu}{(au+b)^{2}}= \displaystyle \frac{b}{a^{2}(au+b)}+ \displaystyle \frac{1}{a^{2}}ln|au+b|+c$
34. $\int \displaystyle \frac{du}{u(au+b)^{2}}= \displaystyle \frac{1}{b(au+b)}+ \displaystyle \frac{1}{b^{2}}ln| \displaystyle \frac{u}{au+b}|+c$
35. $\int (au+b)^{n}du= \displaystyle \frac{(au+b)^{n+1}}{(n+1)a}+c$
36. $\int u(au+b)^{n}du= \displaystyle \frac{(au+b)^{n+2}}{(n+2)a^{2}}- \displaystyle \frac{b(au+b)^{n+1}}{(n+1)a^{2}}+c$
37. $\int u^{m}(au+b)^{n}du= \begin{Bmatrix} \displaystyle \frac{u^{m+1}(au+b)^{n}}{m+n+1}+ \displaystyle \frac{nb}{m+n+1} \int u^{m}(au+b)^{n-1}du \\ \displaystyle \frac{u^{m}(au+b)^{n+1}}{(m+n+1)a}- \displaystyle \frac{mb}{(m+n+1)a} \int u^{n-1}(au+b)^{n}du \\ \displaystyle \frac{-u^{m+1}(au+b)^{n+1}}{(n+1)b}+ \displaystyle \frac{(m+n+2)}{(n+1)b} \int u^{m}(au+b)^{n+1}du \end{Bmatrix}$
ទំរង់ជាប់

$\sqrt{au+b}$

38. $\int \displaystyle \frac{du}{ \sqrt{au+b}}= \displaystyle \frac{2 \sqrt{au+b}}{a}+c$
39. $\int \displaystyle \frac{udu}{ \sqrt{au+b}}= \displaystyle \frac{2(au-2b)}{3a^{2}} \sqrt{au+b}+c$
40. $\int \displaystyle \frac{du}{u \sqrt{au+b}}= \begin{Bmatrix} \displaystyle \frac{1}{ \sqrt{b}}ln| \displaystyle \frac{ \sqrt{au+b}- \sqrt{b}}{ \sqrt{au+b}+ \sqrt{b}}|+c,b>0 \\ \displaystyle \frac{2}{ \sqrt{-b}}arctg \sqrt{ \displaystyle \frac{au+b}{-b}}+c,b<0 \end{Bmatrix}$
41. $\int \sqrt{au+b}du= \displaystyle \frac{2 \sqrt{(au+b)^{3}}}{3a}+c$
42. $\int u \sqrt{au+b}du= \displaystyle \frac{2(3au-2b)}{15a^{2}} \sqrt{(au+b)^{3}}+c$
43. $\int \displaystyle \frac{ \sqrt{au+b}}{u}du=2 \sqrt{au+b}+b \int \displaystyle \frac{du}{u \sqrt{au+b}}$
រូបមន្តជាប់ $u^{2}+a^{2}$

44. $\int \displaystyle \frac{du}{u^{2}+a^{2}}= \displaystyle \frac{1}{a}arctg \displaystyle \frac{u}{a}+c$
45. $\int \displaystyle \frac{udu}{u^{2}+a^{2}}= \displaystyle \frac{1}{2}ln(u^{2}+a^{2})+c$
46. $\int \displaystyle \frac{u^{2}du}{u^{2}+a^{2}}=u-a arctg \displaystyle \frac{u}{a}+c$
47. $\int \displaystyle \frac{du}{u(u^{2}+a^{2})}= \displaystyle \frac{1}{2a^{2}}ln( \displaystyle \frac{u^{2}}{u^{2}+a^{2}})+c$
48. $\int \displaystyle \frac{du}{u^{2}(u^{2}+a^{2})}=- \displaystyle \frac{1}{a^{2}u}- \displaystyle \frac{1}{a^{3}}arctg \displaystyle \frac{u}{a}+c$
49. $\int \displaystyle \frac{du}{(u^{2}+a^{2})^{n}}= \displaystyle \frac{u}{2(n-1)a^{2}(u^{2}+a^{2})^{n-1}}+ \displaystyle \frac{2n-3}{(2n-2)a^{2}} \int \displaystyle \frac{du}{(u^{2}+a^{2})^{n-1}}$
50. $\int \displaystyle \frac{udu}{(u^{2}+a^{2})^{n}}= \displaystyle \frac{-1}{2(n-1)(u^{2}+a^{2})^{n-1}}+c$
51. $\int \displaystyle \frac{du}{u(u^{2}+a^{2})^{n}}= \displaystyle \frac{1}{2(n-1)a^{2}(u^{2}+a^{2})^{n-1}}+ \displaystyle \frac{1}{a^{2}} \int \displaystyle \frac{du}{u(u^{2}+a^{2})^{n-1}}+c$ រូបមន្តជាប់ $u^{2}-a^{2},u^{2}>a^{2}$

52. $\int \displaystyle \frac{du}{u^{2}-a^{2}}= \displaystyle \frac{1}{2a}ln| \displaystyle \frac{u-a}{u+a}|+c$
53. $\int \displaystyle \frac{udu}{u^{2}-a^{2}}= \displaystyle \frac{1}{2}ln(u^{2}-a^{2})+c$
54. $\int \displaystyle \frac{u^{2}du}{u^{2}-a^{2}}=u+ \displaystyle \frac{a}{2}ln| \displaystyle \frac{u-a}{u+a}|+c$
55. $\int \displaystyle \frac{du}{u(u^{2}-a^{2})}= \displaystyle \frac{1}{2a^{2}}ln| \displaystyle \frac{u^{2}-a^{2}}{u^{2}}|+c$
56. $\int \displaystyle \frac{du}{u^{2}(u^{2}-a^{2})}= \displaystyle \frac{1}{a^{2}u}+ \displaystyle \frac{1}{2a^{3}} ln| \displaystyle \frac{u-a}{u+a}|+c$
57. $\int \displaystyle \frac{du}{(u^{2}-a^{2})^{2}}= \displaystyle \frac{-u}{2a^{2}(u^{2}-a^{2})}- \displaystyle \frac{1}{4a^{3}}ln| \displaystyle \frac{u-a}{u+a}|+c$
58. $\int \displaystyle \frac{du}{(u^{2}-a^{2})^{n}}= \displaystyle \frac{-u}{2(n-1)a^{2}(u^{2}-a^{2})^{n-1}}- \displaystyle \frac{2n-3}{(2n-2)a^{2}} \int \displaystyle \frac{du}{(u^{2}-a^{2})^{n-1}}$
59. $\int \displaystyle \frac{udu}{(u^{2}-a^{2})^{n}}= \displaystyle \frac{-1}{2(n-1)a^{2}(u^{2}-a^{2})^{n-1}}+c$
60. $\int \displaystyle \frac{du}{u(u^{2}-a^{2})^{n}}= \displaystyle \frac{-1}{2(n-1)a^{2}(u^{2}-a^{2})^{n-1}}- \displaystyle \frac{1}{a^{2}} \int \displaystyle \frac{du}{u(u^{2}-a^{2})^{n-1}}$ រូបមន្តជាប់ $a^{2}-u^{2},u^{2}<a^{2}$

61. $\int \displaystyle \frac{du}{a^{2}-u^{2}}= \displaystyle \frac{1}{2a}ln| \displaystyle \frac{a+u}{a-u}|+c$
62. $\int \displaystyle \frac{udu}{a^{2}-u^{2}}=- \displaystyle \frac{1}{2}ln|a^{2}-u^{2}|+c$
63. $\int \displaystyle \frac{u^{2}du}{a^{2}-u^{2}}=-u+ \displaystyle \frac{a}{2}ln| \displaystyle \frac{a+u}{a-u}|+c$
64. $\int \displaystyle \frac{du}{u(a^{2}-u^{2})}= \displaystyle \frac{1}{2a^{2}}ln| \displaystyle \frac{u^{2}}{a^{2}-u^{2}}|+c$
65. $\int \displaystyle \frac{du}{(a^{2}-u^{2})^{2}}= \displaystyle \frac{u}{2a^{2}(a^{2}-u^{2})}+ \displaystyle \frac{1}{4a^{3}}ln| \displaystyle \frac{a+u}{a-u}|+c$
66. $\int \displaystyle \frac{udu}{(a^{2}-u^{2})^{2}}= \displaystyle \frac{1}{2(a^{2}-u^{2})}+c$
រូបមន្តជាប់ $\sqrt{u^{2}+a^{2}}$

67. $\int \displaystyle \frac{du}{ \sqrt{u^{2}+a^{2}}}=ln(u+ \sqrt{u^{2}+a^{2}})+c=arcsin \displaystyle \frac{u}{|a|}+c$
68. $\int \displaystyle \frac{udu}{ \sqrt{u^{2}+a^{2}}}= \sqrt{u^{2}+a^{2}}+c$
69. $\int \displaystyle \frac{u^{2}du}{ \sqrt{u^{2}+a^{2}}}= \displaystyle \frac{u \sqrt{u^{2}+a^{2}}}{2}- \displaystyle \frac{a^{2}}{2}ln(u+ \sqrt{u^{2}+a^{2}})+c$
70. $\int \displaystyle \frac{du}{u \sqrt{u^{2}+a^{2}}}=- \displaystyle \frac{1}{a}ln| \displaystyle \frac{a+ \sqrt{u^{2}+a^{2}}}{u}|+c$
71. $\int \sqrt{u^{2}+a^{2}}du= \displaystyle \frac{u \sqrt{u^{2}+a^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}ln(u+ \sqrt{u^{2}+a^{2}})+c$
72. $\int u \sqrt{u^{2}+a^{2}}du= \displaystyle \frac{(u^{2}+a^{2})^{ \frac{3}{2}}}{3}+c$
73. $\int u^{2} \sqrt{u^{2}+a^{2}}du= \displaystyle \frac{u(u^{2}+a^{2})^{ \frac{3}{2}}}{4}- \displaystyle \frac{a^{2}u \sqrt{u^{2}+a^{2}}}{8}- \displaystyle \frac{a^{4}}{8}ln(u+ \sqrt{u^{2}+a^{2}})+c$
74. $\int \displaystyle \frac{ \sqrt{u^{2}+a^{2}}}{u}du= \sqrt{u^{2}+a^{2}}-aln| \displaystyle \frac{a+ \sqrt{u^{2}+a^{2}}}{u}|+c$
75. $\int \displaystyle \frac{ \sqrt{u^{2}+a^{2}}}{u^{2}}du=- \displaystyle \frac{ \sqrt{u^{2}+a^{2}}}{u}+ln(u+ \sqrt{u^{2}+a^{2}})+c$
រូបមន្តជាប់ $\sqrt{u^{2}-a^{2}}$

76. $\int \displaystyle \frac{du}{ \sqrt{u^{2}-a^{2}}}=ln|u+ \sqrt{u^{2}-a^{2}|}+c$
77. $\int \displaystyle \frac{udu}{ \sqrt{u^{2}-a^{2}}}= \sqrt{u^{2}-a^{2}}+c$
78. $\int \displaystyle \frac{u^{2}du}{ \sqrt{u^{2}-a^{2}}}= \displaystyle \frac{u \sqrt{u^{2}-a^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}ln|u+ \sqrt{u^{2}-a^{2}}|+c$
79. $\int \displaystyle \frac{du}{u \sqrt{u^{2}-a^{2}}}= \displaystyle \frac{1}{|a|}arcsec| \displaystyle \frac{u}{a}|+c$
80. $\int \sqrt{u^{2}-a^{2}}du= \displaystyle \frac{u \sqrt{u^{2}-a^{2}}}{2}- \displaystyle \frac{a^{2}}{2}ln|u+ \sqrt{u^{2}-a^{2}}|+c$
81. $\int u \sqrt{u^{2}-a^{2}}du= \displaystyle \frac{(u^{2}-a^{2})^{ \frac{3}{2}}}{3}+c$
82. $\int u^{2} \sqrt{u^{2}-a^{2}}du= \displaystyle \frac{u(u^{2}-a^{2})^{ \frac{3}{2}}}{4}+ \displaystyle \frac{a^{2}u \sqrt{u^{2}-a^{2}}}{8}- \displaystyle \frac{a^{4}}{8}ln|u+ \sqrt{u^{2}-a^{2}}|+c$
83. $\int \displaystyle \frac{ \sqrt{u^{2}-a^{2}}}{u}du= \sqrt{u^{2}-a^{2}}-|a|arcsec| \displaystyle \frac{u}{a}|+c$
84. $\int \displaystyle \frac{ \sqrt{u^{2}-a^{2}}}{u^{2}}du=- \displaystyle \frac{ \sqrt{u^{2}-a^{2}}}{u}+ln|u+ \sqrt{u^{2}-a^{2}}|+c$
85. $\int \displaystyle \frac{du}{(u^{2}-a^{2})^{ \frac{3}{2}}}=- \displaystyle \frac{u}{a^{2} \sqrt{u^{2}-a^{2}}}+c$
រូបមន្តជាប់ $\sqrt{a^{2}-u^{2}}$

86. $\int \displaystyle \frac{du}{ \sqrt{a^{2}-u^{2}}}=arcsin \displaystyle \frac{u}{|a|}+c$

87. $\int \displaystyle \frac{udu}{ \sqrt{a^{2}-u^{2}}}=- \sqrt{a^{2}-u^{2}}+c$

88. $\int \displaystyle \frac{u^{2}du}{ \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{u \sqrt{a^{2}-u^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}arcsin \displaystyle \frac{u}{|a|}+c$

89. $\int \displaystyle \frac{du}{u \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{1}{a}ln| \displaystyle \frac{a+ \sqrt{a^{2}-u^{2}}}{u}|+c$

90. $\int \displaystyle \frac{du}{u^{2} \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{ \sqrt{a^{2}-u^{2}}}{a^{2}u}+c$

91. $\int \sqrt{a^{2}-u^{2}}du= \displaystyle \frac{u \sqrt{a^{2}-u^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}arcsin \displaystyle \frac{u}{|a|}+c$

92. $\int u \sqrt{a^{2}-u^{2}}du=- \displaystyle \frac{(a^{2}-u^{2})^{ \frac{3}{2}}}{3}+c$

93. $\int u^{2} \sqrt{a^{2}-u^{2}}du=- \displaystyle \frac{u(a^{2}-u^{2})^{ \frac{3}{2}}}{4}+ \displaystyle \frac{a^{2}u \sqrt{a^{2}-u^{2}}}{8}+ \displaystyle \frac{a^{4}}{8}arcsin \displaystyle \frac{u}{|a|}+c$

94. $\int \displaystyle \frac{ \sqrt{a^{2}-u^{2}}}{u}du= \sqrt{a^{2}-u^{2}}-aln| \displaystyle \frac{a+ \sqrt{a^{2}-u^{2}}}{u}|+c$

អាំងតេក្រាលជាប់អនុគមន៍ត្រីកោណមាត្រ
96. $\int sinudu=- \displaystyle \frac{cosau}{a}+c$
97. $\int u sinaudu= \displaystyle \frac{sinau}{a^{2}}- \frac{ucosau}{a}+c$
98. $\int u^{2}sinaudu= \displaystyle \frac{2u}{a^{2}}sinau+( \frac{2}{a^{3}}- \frac{u^{2}}{a})cosau+c$
99. $\int \displaystyle \frac{du}{sinau}= \frac{1}{a}ln(cscau-cotau)= \frac{1}{a}ln|tan \frac{au}{2}|+c$
100. $\int sin^{2}audu= \displaystyle \frac{u}{2}- \frac{sin2au}{4a}+c$
101. $\int usin^{2}audu= \displaystyle \frac{u^{2}}{4}- \frac{usin2au}{4a}- \frac{cos2au}{8a^{2}}+c$
102. $\int \displaystyle \frac{du}{sin^{2}au}=- \displaystyle \frac{1}{a}cotau+c$
103. $\int sinpusinqudu= \displaystyle \frac{sin(p-q)u}{2(p-q)}- \frac{sin(p+q)u}{2(p+q)}+c$
104. $\int \displaystyle \frac{du}{1-sinau}= \displaystyle \frac{1}{a}tan( \frac{ \pi}{4}+ \frac{au}{2})+c$
105. $\int \displaystyle \frac{udu}{1-sinau}= \displaystyle \frac{u}{a}tan( \frac{ \pi}{4}+ \frac{au}{2})+ \frac{2}{a^{2}}ln|sin( \frac{ \pi}{4}- \frac{au}{2})|+c$
106. $\int \displaystyle \frac{du}{1+sinau}=- \displaystyle \frac{1}{a}tan( \frac{ \pi}{4}- \frac{au}{2})+c$
107. $\int \displaystyle \frac{du}{p+qsinau}=- \begin{Bmatrix} \displaystyle \frac{2}{a \sqrt{p^{2}-q^{2}}}arctg \displaystyle \frac{ptan \frac{1}{2}au+q}{ \sqrt{p^{2}-q^{2}}}+c \\ \displaystyle \frac{1}{a \sqrt{p^{2}-q^{2}}}ln| \displaystyle \frac{ptan \frac{1}{2}au+q- \sqrt{p^{2}-q^{2}}}{ptan \frac{1}{2}au+q+ \sqrt{p^{2}-q^{2}}}|+c \end{Bmatrix}$
108. $\int u^{m}sinaudu=- \displaystyle \frac{u^{m}cosau}{a}+ \frac{mu^{m-1}sinau}{a^{2}}- \frac{m(m-1)}{a^{2}} \int u^{m-2}sinaudu$
109. $\int sin^{n}audu=- \displaystyle \frac{sin^{n-1}aucosau}{an}+ \frac{n-1}{n} \int sin^{n-2}audu$
110. $\int \displaystyle \frac{du}{sin^{n}au}= \frac{-cosau}{a(n-1)sin^{n-1}au}+ \frac{n-2}{n-1} \int \frac{du}{sin^{n-2}au}$
111. $\int cosaudu= \displaystyle \frac{sinau}{a}+c$
112. $\int ucosaudu= \displaystyle \frac{cosau}{a^{2}}+ \frac{usinau}{a}+c$
113. $\int u^{2}cosaudu= \displaystyle \frac{2u}{a}cosau+( \frac{u^{2}}{a}- \frac{2}{a^{3}})sinau+c$
114. $\int \displaystyle \frac{du}{coau}= \frac{1}{a}ln|secau+tanau|= \frac{1}{a}ln|tan( \frac{ \pi}{4}+ \frac{au}{2})|+c$
115. $\int cos^{2}audu= \displaystyle \frac{u}{2}+ \frac{sin2au}{4a}+c$
116. $\int ucos^{2}audu= \displaystyle \frac{u^{2}}{4}+ \frac{usin2au}{4a}+ \frac{cos2ar}{8a^{2}}+c$
117. $\int \displaystyle \frac{du}{cos^{2}au}= \frac{tanau}{a}+c$
118. $\int cosqucospudu= \displaystyle \frac{sin(q-p)u}{2(q-p)}+ \frac{sin(q+p)u}{2(q+p)}+c$
119. $\int \displaystyle \frac{du}{p+qcosau}= \begin{Bmatrix} \displaystyle \frac{2}{a \sqrt{p^{2}-q^{2}}}arctg[ \sqrt{(p-q)/(p+q)}tan \frac{1}{2}au]+c,|p|>|q| \\ \displaystyle \frac{1}{a \sqrt{q^{2}-p^{2}}}ln[ \frac{tan \frac{1}{2}au+ \sqrt{(q+p)/(q-p)}}{tan \frac{1}{2}au- \sqrt{(q+p)/q-p)}}]+c,|p|<|q| \end{Bmatrix}$
120. $\int u^{m}cosaudu= \displaystyle \frac{u^{m}sinau}{a}+ \frac{mu^{m-1}}{a^{2}}cosau- \frac{m(m-1)}{a^{2}} \int u^{m-2}cosaudu$
121. $\int cos^{n}audu= \displaystyle \frac{sinaucos^{n-1}au}{an}+ \frac{n-1}{n} \int cos^{n-2}audu$
122. $\int \displaystyle \frac{du}{cos^{n}au}= \frac{sinau}{a(n-1)cos^{n-1}au}+ \frac{n-2}{n-1} \int \frac{du}{cos^{n-2}au}$
123. $\int sinau cosaudu= \displaystyle \frac{sin^{2}au}{2a}+c$
124. $\int sinpucosqudu=- \displaystyle \frac{cos(p-q)u}{2(p-q)}- \frac{cos(p+q)u}{2(p+q)}+c$
125. $\int sin^{n}aucosaudu= \displaystyle \frac{sin^{n+1}au}{(n+1)a}+c$
126. $\int cos^{n}ausinaudu=- \displaystyle \frac{cos^{n+1}au}{(n+1)a}+c$
127. $\int sin^{2}aucos^{2}audu= \displaystyle \frac{u}{8}- \frac{sin4au}{32a}+c$
128. $\int \displaystyle \frac{du}{sinaucosau}= \frac{1}{a}ln|tanau|+c$
129. $\int \displaystyle \frac{du}{cosau(1+sinau}= \frac{1}{2a(1+sinau)}+ \frac{1}{2a}ln|tan( \frac{au}{2}+ \frac{ \pi}{4})|+c$
130. $\int \displaystyle \frac{du}{sinau(1 \pm cosau)}= \pm \frac{1}{2a(1 \pm cosau)}+ \frac{1}{2a}ln|tan \frac{au}{2}|+c$
131. $\int \displaystyle \frac{du}{sinau \pm cosau}= \frac{1}{a \sqrt{2}}ln|tan( \frac{au}{2} \pm \frac{ \pi}{8})|+c$
132. $\int \displaystyle \frac{sinaudu}{sinau \pm cosau}= \frac{u}{2} \mp \frac{1}{2a}ln|sinau \pm cosau|+c$
133. $\int \displaystyle \frac{cosaudu}{sinau \pm cosau}= \pm [ \frac{u}{2}+ \frac{1}{2a}ln|sinau \pm cosau|]+c$
134. $\int \displaystyle \frac{sinaudu}{p+qcosau}=- \frac{1}{aq}ln|p+qcosau|+c$
135. $\int \displaystyle \frac{cosaudu}{p+qsinau}= \frac{1}{aq}ln|p+qsinau|+c$
136. $\int sin^{m}aucos^{n}audu=\displaystyle \begin{Bmatrix} - \displaystyle \frac{sin^{m-1}aucos^{n+1}au}{a(n+m)}+ \frac{m-1}{m+n} \int sin^{m-2}aucos^{n}audu \\ \displaystyle \frac{sin^{m+1}aucos^{n-1}au}{a(m+n)}+ \frac{n-1}{m+n} \int sin^{m}aucos^{n-2}audu \end{Bmatrix}$
137. $\int tanaudu=- \displaystyle \frac{1}{a}ln|cosau|= \frac{1}{a}ln|secau|+c$
138. $\int tan^{2}audu= \displaystyle \frac{tanau}{a}-u+c$
139. $\int tan^{n}ausec^{2}audu= \displaystyle \frac{tan^{n-1}au}{(n+1)a}+c$
140. $\int tan^{n}audu= \displaystyle \frac{tan^{n-1}au}{(a-1)a}- \int tan^{n-2}audu$
141. $\int cotaudu= \displaystyle \frac{1}{a}ln|sinau|+c$
142. $\int cot^{2}audu=- \displaystyle \frac{cotau}{a}-u+c$
143. $\int cot^{n}aucsc^{2}audu=- \displaystyle \frac{cot^{n-1}au}{(n+1)a}+c$
144. $\int cot^{n}audu=- \displaystyle \frac{cot^{n-1}au}{(n-1)a}- \int cot^{n-2}audu$
145. $\int sec^{2}audu= \displaystyle \frac{tanau}{a}+c$
146. $\int secaudu= \displaystyle \frac{1}{a}ln|secau+tanau|= \frac{1}{a}ln|tan( \frac{au}{2}+ \frac{ \pi}{4})|+c$
147. $\int sec^{3}audu= \displaystyle \frac{secautanau}{2a}+ \frac{1}{2a}ln|secau+tanau|+c$
148. $\int sec^{n}autanaudu= \displaystyle \frac{sec^{n}au}{na}+c$
149. $\int sec^{n}audu= \displaystyle \frac{sec^{n-2}autanau}{a(n-1)}+ \frac{n-2}{n-1} \int sec^{n-2}audu$
150. $\int cscaudu= \displaystyle \frac{1}{a}ln|cscau-cotau|= \frac{1}{a}ln|tan \frac{au}{2}|+c$
151. $\int csc^{2}audu=- \displaystyle \frac{cotau}{a}+c$
152. $\int csc^{n}aucotaudu=- \displaystyle \frac{csc^{n}au}{an}+c$
អាំងតេក្រាលជាប់អនុគមន៍ច្រាសត្រីកោណមាត្រ

154. $\int arcsin \displaystyle \frac{u}{a}du=uarcsin \frac{a}{u}+ \sqrt{a^{2}-u^{2}}+c$
155. $\int uarcsin \displaystyle \frac{u}{a}du=( \frac{u^{2}}{2}- \frac{a^{2}}{4})arcsin \frac{u}{a}+ \frac{u \sqrt{a^{2}-u^{2}}}{4}+c$
156. $\int arccos \displaystyle \frac{u}{a}du=uarccos \frac{u}{a}- \sqrt{a^{2}-u^{2}}+c$
157. $\int uarccos \displaystyle \frac{u}{a}du=( \frac{u^{2}}{2}- \frac{a^{2}}{4})arccos \frac{u}{a}- \frac{u \sqrt{a^{2}-u^{2}}}{4}+c$
158. $\int arctg \displaystyle \frac{u}{a}du=uacrtg \frac{u}{a}- \frac{a}{2}ln(u^{2}+a^{2})+c$
159. $\int uarctg \displaystyle \frac{u}{a}du= \frac{1}{2}(u^{2}+a^{2})acrtg \frac{u}{a}- \frac{au}{2}+c$
160. $\int u^{m}acrsin \displaystyle \frac{u}{a}du= \frac{u^{m+1}}{m+1}arcsin \frac{u}{a}- \frac{1}{m+1} \int \frac{u^{m+1}}{ \sqrt{a^{2}-u^{2}}}du$
161. $\int u^{m}arccos \displaystyle \frac{u}{a}du= \frac{u^{m+1}}{m+1}arccos \frac{u}{a}+ \frac{1}{m+1} \int \frac{u^{m+1}}{ \sqrt{a^{2}-u^{2}}}du$
162. $\int u^{m}arctg \displaystyle \frac{u}{a}du= \frac{u^{m+1}}{m+1}arctg \frac{u}{a}- \frac{a}{m+1} \int \frac{u^{m+1}}{u^{2}+a^{2}}du$
អាំងតេក្រាលជាប់ $e^{au}$

163. $\int e^{au}du= \displaystyle \frac{e^{au}}{a}+c$
164. $\int ue^{au}du= \displaystyle \frac{e^{au}}{a}(u- \frac{1}{a}+c$
165. $\int u^{2}e^{au}du= \displaystyle \frac{e^{au}}{a}(u^{2}- \frac{2u}{a}+ \frac{2}{a^{2}}+c$
166. $\int u^{n}e^{au}du= \displaystyle \frac{u^{n}e^{au}}{a}- \frac{n}{a} \int u^{n-1}e^{au}du= \frac{e^{au}}{a}[u^{n}- \frac{nu^{n-1}}{a}+ \frac{n(n-1)u^{n-2}}{a^{2}}-...+ \frac{(-1)^{n}n!}{a^{n}}],$ បើn ជាចំនួនគត់វិជ្ជមាន
167. $\int \displaystyle \frac{du}{p-qe^{au}}= \frac{u}{p}- \frac{1}{ap}ln|p+qe^{au}|+c$
168. $\int e^{au}sinbudu= \displaystyle \frac{e^{au}(asinbu-bcosbu)}{a^{2}+b^{2}}+c$
169. $\int e^{au}cosbudu= \displaystyle \frac{e^{au}(acosbu+bsinbu)}{a^{2}+b^{2}}+c$
170. $\int ue^{au}sinbudu= \displaystyle \frac{ue^{au}(asinbu-bcosbu)}{a^{2}+b^{2}}- \frac{e^{au}[(a^{2}-b^{2})sinbu-2abcosbu]}{(a^{2}-b^{2})^{2}}+c$
171. $\int ue^{au}cosbudu= \displaystyle \frac{ue^{au}(acosbu+bsinbu)}{a^{2}+b^{2}}- \frac{e^{au}[(a^{2}-b^{2})cosbu+2absinbu]}{(a^{2}-b^{2})^{2}}+c$
172. $\int e^{au}sin^{n}budu= \displaystyle \frac{e^{au}sin^{n-1}bu}{a^{2}+n^{2}b^{2}}(asinbu-nbcosbu)+ \frac{n(n-1)b^{2}}{a^{2}+n^{2}b^{2}} \int e^{au}sin^{n-2}budu$
173. $\int e^{au}cos^{n}budu= \displaystyle \frac{e^{au}cos^{n-1}bu}{a^{2}+n^{2}b^{2}}(acosbu+nbsinbu)+ \frac{n(n-1)b^{2}}{a^{2}+n^{2}b^{2}} \int e^{au}cos^{n-2}budu$
អាំងតេក្រាលជាប់ lnu

174. $\int lnudu=ulnu-u+c$
175. $\int ulnudu= \displaystyle \frac{u^{2}}{2}(lnu- \frac{1}{2})+c$
176. $\int u^{m}lnudu= \displaystyle \frac{u^{m+1}}{m+1}(lnu- \frac{1}{m+1}$
177. $\int \displaystyle \frac{lnu}{u}du= \frac{1}{2}ln^{2}u+c$
178. $\int \displaystyle \frac{ln^{n}udu}{u}= \frac{ln^{n+1}u}{n+1}+c$ , បើ $n \neq -1$
179. $\int \displaystyle \frac{du}{ulnu}=ln|lnu|+c$
180. $\int ln^{n}udu=uln^{n}u-n \int ln^{n-1}udu, n \neq -1$
181. $\int u^{m}ln^{n}udu= \displaystyle \frac{u^{m+1}ln^{n}u}{m+1}- \frac{n}{m+1} \int u^{m}ln{n-1}udu,(m,n \neq -1)$
182. $\int ln(u^{2}+a^{2})du=uln(u^{2}+a^{2})-2u+2aarctg \frac{u}{a}+c$
183. $\int ln|u^{2}+a^{2}|du=uln|u^{2}+a^{2}|-su+aln| \dfrac{u+a}{u-a}|+c$
អាំងតេក្រាលជាប់អនុគមន៍អីប៉ែបូល

184. $\int sinhaudu= \dfrac{coshau}{a}+c$
185. $\int usinhaudu= \dfrac{ucoshau}{a}- \dfrac{sinhau}{a^{2}}+c$
186. $\int \dfrac{sinhau}{a}+c$
187. $\int ucoshaudu= \dfrac{usinhau}{a}- \dfrac{coshau}{a^{2}}+c$
188. $\int cosh^{2}audu= \dfrac{u}{2}+ \dfrac{sinhaucoshau}{2a}+c$
189. $\int sinh^{2}audu= \dfrac{sinhaucoshau}{2a}- \dfrac{u}{2}+c$
190. $\int sinh^{n}audu= \dfrac{sinh^{n-1}aucoshau}{an}- \dfrac{n-1}{n} \int sinh^{n-2}audu$
191. $\int cosh^{n}audu= \dfrac{cosh^{n-1}ausinhau}{an}- \dfrac{n-1}{n} \int cosh^{n-2}audu$
192. $\int sinhaucoshaudu= \dfrac{sinh^{2}au}{2a}+c$
193. $\int sinhpu coshqudu= \dfrac{cosh(p+q)u}{2(p+q)}+ \dfrac{cosh(p-q)u}{2(p-q)}+c$
194. $\int tanhaudu= \dfrac{1}{a}lncoshau+c$
195. $\int tanh^{2}audu=u- \dfrac{tanhau}{a}+c$
196. $\int tanh^{n}audu= \dfrac{-tanh^{n-1}au}{a(n-1)}+ \int tanh^{n-2}audu$
197. $\int cothaudu= \dfrac{1}{a}ln|sinhau|+c$
198. $\int coth^{2}audu=u- \dfrac{cothau}{a}+c$
199. $\int sechaudu= \dfrac{2}{a}arctge^{au}+c$
200. $\int sech^{2}audu= \dfrac{tanhau}{a}$
201. $\int sech^{n}audu= \dfrac{sech^{n-2}autanhau}{a(n-1)}+ \dfrac{n-2}{n-1} \int sech^{n-2}audu$
202. $\int cschaudu= \dfrac{1}{a}ln|tanh \dfrac{au}{2}|+c$
203. $\int csch^{2}audu=- \dfrac{cothau}{a}+c$
204. $\int sechu tanhudu= -sechu+c$
205. $\int cschu coshudu=-cschu+c$
អាំងតេក្រាលកំនត់មួយចំនួន

206. $\int_{0}^{ \infty} \dfrac{dx}{x^{2}+a^{2}}= \dfrac{ \pi}{2a}$
207. $\int_{0}^{ \infty} \dfrac{x^{p-1}}{1+x}dx= \dfrac{ \pi}{sinp \pi}$
208. $\int_{0}^{a} \dfrac{dx}{ \sqrt{a^{2}-x^{2}}}= \dfrac{ \pi}{2}$
209. $\int_{0}^{a} sqrt{a^{2}-x^{2}}dx= \dfrac{ \pi a^{2}}{4}$
210. $\int_{0}^{ \pi} sinmxsinnxdx= \begin{Bmatrix} 0, ( m,n, integers , m \neq n) \\ \dfrac{ \pi}{2}, ( m,n, integers , m=n) \end{Bmatrix}$
211. $\int_{0}^{ \pi} cosmxcosnxdx= \begin{Bmatrix} o, ( m,n, integers , m \neq n) \\ \dfrac{ \pi}{2}, ( m,n, integers ,m=n) \end{Bmatrix}$
212. $\int_{0}^{ \pi} sinmxcosnxdx= \begin{Bmatrix} o,( m,n,intergers ,m+n = even) \\ \dfrac{2m}{(m^{2}-n^{2}},(if m,n, integers , m+n = odd) \end{Bmatrix}$
213. $\int_{o}^{ \frac{ \pi}{2}} sin^{2}xdx= \int_{0}^{ \frac{ \pi}{2}} cos^{2}xdx= \dfrac{ \pi}{4}$
214. $\int_{0}^{ \infty} e^{-ax}cosbxdx= \dfrac{a}{a^{2}+b^{2}}$
215. $\int_{0}^{ \infty} e^{-ax}sinbxdx= \dfrac{b}{a^{2}+b^{2}}$
216. $\int_{0}^{ \infty} e^{-a^{2}x^{2}}dx= \dfrac{ \sqrt{ \pi}}{2a}$
217. $\int_{0}^{ \frac{ \pi}{2}} sin^{2m}xdx= \int_{0}^{ \frac{ \pi}{2}} cos^{2m}xdx= \dfrac{1.3.5.....(2m-1)}{2.4.6.....2m} \dfrac{ \pi}{2}, m=1,2,3....$
218. $\int_{0}^{ \frac{ \pi}{2}} sin^{2m+1}xdx= \int_{0}^{ \frac{ \pi}{2}} cos^{2m+1}xdx= \dfrac{2.4.6.....2m}{1.3.5.....(2m+1)}, m=1,2,3,....$
219. $\int_{0}^{ \infty} \dfrac{e^{-x}}{ \sqrt{x}}dx= \sqrt{ \pi}$
220. $\int_{0}^{1} x^{m}(lnx)^{n}dx= \dfrac{(-1)^{n}n!}{(m+1)^{n+1}}, m \neq -1$

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