انتگرال‌های پایه‌ای

1. $$\int x^n\,dx = \frac{1}{n + 1} x^{n + 1},\quad n \neq - 1$$
2. $$\int \frac{1}{x}\,dx = \ln | x |$$
3. $$\int u\,dv = u v - \int v\,du$$
4. $$\int \frac{1}{a x + b}\,dx = \frac{1}{a} \ln | a x + b |$$

انتگرال‌های کسری

5. $$\int \frac{1}{{( x + a )}^2}\,dx = - \frac{1}{x + a}$$
6. $$\int {( x + a )}^n\,dx = \frac{{( x + a )}^{n + 1}}{n + 1} , n \neq - 1$$
7. $$\int x {( x + a )}^n\,dx = \frac{{( x + a )}^{n + 1} ( ( n + 1 ) x - a )}{( n + 1 ) ( n + 2 )}$$
8. $$\int \frac{1}{1 + x^2}\,dx = \tan^{- 1} x$$
9. $$\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a} \tan^{- 1} \frac{x}{a}$$
10. $$\int \frac{x}{a^2 + x^2}\,dx = \frac{1}{2} \ln | a^2 + x^2 |$$
11. $$\int \frac{x^2}{a^2 + x^2}\,dx = x - a \tan^{- 1} \frac{x}{a}$$
12. $$\int \frac{x^3}{a^2 + x^2}\,dx = \frac{1}{2} x^2 - \frac{1}{2} a^2 \ln | a^2 + x^2 |$$
13. $$\int \frac{1}{a x^2 + b x + c}\,dx = \frac{2}{\sqrt{4 a c - b^2}} \tan^{- 1} \frac{2 a x + b}{\sqrt{4 a c - b^2}}$$
14. $$\int \frac{1}{( x + a ) ( x + b )}\,dx = \frac{1}{b - a} \ln \frac{a + x}{b + x} , a \neq b$$
15. $$\int \frac{x}{{( x + a )}^2}\,dx = \frac{a}{a + x} + \ln | a + x |$$
16. $$\int \frac{x}{a x^2 + b x + c}\,dx = \frac{1}{2 a} \ln | a x^2 + b x + c | - \frac{b}{a \sqrt{4 a c - b^2}} \tan^{- 1} \frac{2 a x + b}{\sqrt{4 a c - b^2}}$$

انتگرال‌های رادیکالی

17. $$\int \sqrt{x - a}\,dx = \frac{2}{3} {( x - a )}^{3 / 2}$$
18. $$\int \frac{1}{\sqrt{x \pm a}}\,dx = 2 \sqrt{x \pm a}$$
19. $$\int \frac{1}{\sqrt{a - x}}\,dx = - 2 \sqrt{a - x}$$
20. $$\int x \sqrt{x - a}\,dx = \left\{ \begin{array}{l} \displaystyle \frac{2 a}{3} {\left( x - a \right)}^{3 / 2} + \frac{2}{5} {\left( x - a \right)}^{5 / 2},\quad\text{or}\quad \\ \displaystyle \frac{2}{3} x {( x - a )}^{3 / 2} - \frac{4}{15} {( x - a )}^{5 / 2},\quad\text{or}\quad \\ \displaystyle \frac{2}{15} ( 2 a + 3 x ) {( x - a )}^{3 / 2} \end{array} \right.$$
21. $$\int \sqrt{a x + b}\,dx = \left( \frac{2 b}{3 a} + \frac{2 x}{3} \right) \sqrt{a x + b}$$
22. $$\int {( a x + b )}^{3 / 2}\,dx = \frac{2}{5 a} {( a x + b )}^{5 / 2}$$
23. $$\int \frac{x}{\sqrt{x \pm a}}\,dx = \frac{2}{3} ( x \mp 2 a ) \sqrt{x \pm a}$$
24. $$\int \sqrt{\frac{x}{a - x}}\,dx = - \sqrt{x ( a - x )} - a \tan^{- 1} \frac{\sqrt{x ( a - x )}}{x - a}$$
25. $$\int \sqrt{\frac{x}{a + x}}\,dx = \sqrt{x ( a + x )} - a \ln \left[ \sqrt{x} + \sqrt{x + a} \right]$$
26. $$\int x \sqrt{a x + b}\,dx = \frac{2}{15 a^2} ( - 2 b^2 + a b x + 3 a^2 x^2 ) \sqrt{a x + b}$$
27. $$\int \sqrt{x ( a x + b )}\,dx = \frac{1}{4 a^{3 / 2}} \left[ ( 2 a x + b ) \sqrt{a x ( a x + b )} - b^2 \ln \left| a \sqrt{x} + \sqrt{a ( a x + b )} \right| \right]$$
28. $$\int \sqrt{x^3 ( a x + b )}\,dx = \left[ \frac{b}{12 a} - \frac{b^2}{8 a^2 x} + \frac{x}{3} \right] \sqrt{x^3 ( a x + b )} + \frac{b^3}{8 a^{5 / 2}} \ln \left| a \sqrt{x} + \sqrt{a ( a x + b )} \right|$$
29. $$\int \sqrt{x^2 \pm a^2}\,dx = \frac{1}{2} x \sqrt{x^2 \pm a^2} \pm \frac{1}{2} a^2 \ln \left| x + \sqrt{x^2 \pm a^2} \right|$$
30. $$\int \sqrt{a^2 - x^2}\,dx = \frac{1}{2} x \sqrt{a^2 - x^2} + \frac{1}{2} a^2 \tan^{- 1} \frac{x}{\sqrt{a^2 - x^2}}$$
31. $$\int x \sqrt{x^2 \pm a^2}\,dx = \frac{1}{3} {\left( x^2 \pm a^2 \right)}^{3 / 2}$$
32. $$\int \frac{1}{\sqrt{x^2 \pm a^2}}\,dx = \ln \left| x + \sqrt{x^2 \pm a^2} \right|$$
33. $$\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \sin^{- 1} \frac{x}{a}$$
34. $$\int \frac{x}{\sqrt{x^2 \pm a^2}}\,dx = \sqrt{x^2 \pm a^2}$$
35. $$\int \frac{x}{\sqrt{a^2 - x^2}}\,dx = - \sqrt{a^2 - x^2}$$
36. $$\int \frac{x^2}{\sqrt{x^2 \pm a^2}}\,dx = \frac{1}{2} x \sqrt{x^2 \pm a^2} \mp \frac{1}{2} a^2 \ln \left| x + \sqrt{x^2 \pm a^2} \right|$$
37. $$\int \sqrt{a x^2 + b x + c}\,dx = \frac{b + 2 a x}{4 a} \sqrt{a x^2 + b x + c} + \frac{4 a c - b^2}{8 a^{3 / 2}} \ln \left| 2 a x + b + 2 \sqrt{a ( a x^2 + b x + c )} \right|$$
38. $$\begin{aligned} \int x \sqrt{a x^2 + b x + c}\,dx = \frac{1}{48 a^{5/2}} \Big( & 2 \sqrt{a} \sqrt{a x^2 + b x + c} \left( -3 b^2 + 2 a b x + 8 a (c + a x^2) \right) \\ & + 3 (b^3 - 4 a b c) \ln \left| b + 2 a x + 2 \sqrt{a} \sqrt{a x^2 + b x + c} \right| \Big) \end{aligned}$$
39. $$\int \frac{1}{\sqrt{a x^2 + b x + c}}\,dx = \frac{1}{\sqrt{a}} \ln \left| 2 a x + b + 2 \sqrt{a ( a x^2 + b x + c )} \right|$$
40. $$\int \frac{x}{\sqrt{a x^2 + b x + c}}\,dx = \frac{1}{a} \sqrt{a x^2 + b x + c} - \frac{b}{2 a^{3 / 2}} \ln \left| 2 a x + b + 2 \sqrt{a ( a x^2 + b x + c )} \right|$$
41. $$\int \frac{dx}{{( a^2 + x^2 )}^{3 / 2}} = \frac{x}{a^2 \sqrt{a^2 + x^2}}$$

انتگرال‌های لگاریتمی

42. $$\int \ln a x\,dx = x \ln a x - x$$
43. $$\int x \ln x\,dx = \frac{1}{2} x^2 \ln x - \frac{x^2}{4}$$
44. $$\int x^2 \ln x\,dx = \frac{1}{3} x^3 \ln x - \frac{x^3}{9}$$
45. $$\int x^n \ln x\,dx = x^{n + 1} \left( \frac{\ln x}{n + 1} - \frac{1}{{( n + 1 )}^2} \right),\quad n \neq - 1$$
46. $$\int \frac{\ln a x}{x}\,dx = \frac{1}{2} {\left( \ln a x \right)}^2$$
47. $$\int \frac{\ln x}{x^2}\,dx = - \frac{1}{x} - \frac{\ln x}{x}$$
48. $$\int \ln ( a x + b )\,dx = \left( x + \frac{b}{a} \right) \ln ( a x + b ) - x , a \neq 0$$
49. $$\int \ln ( x^2 + a^2 )\,dx = x \ln ( x^2 + a^2 ) + 2 a \tan^{- 1} \frac{x}{a} - 2 x$$
50. $$\int \ln ( x^2 - a^2 )\,dx = x \ln ( x^2 - a^2 ) + a \ln \frac{x + a}{x - a} - 2 x$$
51. $$\int \ln \left( a x^2 + b x + c \right)\,dx = \frac{1}{a} \sqrt{4 a c - b^2} \tan^{- 1} \frac{2 a x + b}{\sqrt{4 a c - b^2}} - 2 x + \left( \frac{b}{2 a} + x \right) \ln \left( a x^2 + b x + c \right)$$
52. $$\int x \ln ( a x + b )\,dx = \frac{b x}{2 a} - \frac{1}{4} x^2 + \frac{1}{2} \left( x^2 - \frac{b^2}{a^2} \right) \ln ( a x + b )$$
53. $$\int x \ln \left( a^2 - b^2 x^2 \right)\,dx = - \frac{1}{2} x^2 + \frac{1}{2} \left( x^2 - \frac{a^2}{b^2} \right) \ln \left( a^2 - b^2 x^2 \right)$$
54. $$\int {( \ln x )}^2\,dx = 2 x - 2 x \ln x + x {( \ln x )}^2$$
55. $$\int {( \ln x )}^3\,dx = - 6 x + x {( \ln x )}^3 - 3 x {( \ln x )}^2 + 6 x \ln x$$
56. $$\int x {( \ln x )}^2\,dx = \frac{x^2}{4} + \frac{1}{2} x^2 {( \ln x )}^2 - \frac{1}{2} x^2 \ln x$$
57. $$\int x^2 {( \ln x )}^2\,dx = \frac{2 x^3}{27} + \frac{1}{3} x^3 {( \ln x )}^2 - \frac{2}{9} x^3 \ln x$$

انتگرال‌های توانی

58. $$\int e^{a x}\,dx = \frac{1}{a} e^{a x}$$
59. $$\int \sqrt{x} e^{a x}\,dx = \frac{1}{a} \sqrt{x} e^{a x} + \frac{i \sqrt{\pi}}{2 a^{3 / 2}} \operatorname{erf} \left( i \sqrt{a x} \right),\quad\text{where}\quad \operatorname{erf} ( x ) = \frac{2}{\sqrt{\pi}} \int_0^x e^{- t^2}\,dt$$
60. $$\int x e^x\,dx = ( x - 1 ) e^x$$
61. $$\int x e^{a x}\,dx = \left( \frac{x}{a} - \frac{1}{a^2} \right) e^{a x}$$
62. $$\int x^2 e^x\,dx = \left( x^2 - 2 x + 2 \right) e^x$$
63. $$\int x^2 e^{a x}\,dx = \left( \frac{x^2}{a} - \frac{2 x}{a^2} + \frac{2}{a^3} \right) e^{a x}$$
64. $$\int x^3 e^x\,dx = \left( x^3 - 3 x^2 + 6 x - 6 \right) e^x$$
65. $$\int x^n e^{a x}\,dx = \frac{x^n e^{a x}}{a} - \frac{n}{a} \int x^{n - 1} e^{a x} \,dx$$
66. $$\int x^n e^{a x}\,dx = \frac{{( - 1 )}^n}{a^{n + 1}} \Gamma [ 1 + n , - a x ],\quad\text{where}\quad \Gamma ( a , x ) = \int_x^{\infty} t^{a - 1} e^{- t} \,dt$$
67. $$\int e^{a x^2}\,dx = - \frac{i \sqrt{\pi}}{2 \sqrt{a}} \operatorname{erf} \left( i x \sqrt{a} \right)$$
68. $$\int e^{- a x^2}\,dx = \frac{\sqrt{\pi}}{2 \sqrt{a}} \operatorname{erf} \left( x \sqrt{a} \right)$$
69. $$\int x e^{- a x^2}\,dx = - \frac{1}{2 a} e^{- a x^2}$$
70. $$\int x^2 e^{- a x^2}\,dx = \frac{1}{4} \sqrt{\frac{\pi}{a^3}} \operatorname{erf} ( x \sqrt{a} ) - \frac{x}{2 a} e^{- a x^2}$$

انتگرال‌های مثلثاتی

71. $$\int \sin a x\,dx = - \frac{1}{a} \cos a x$$
72. $$\int \sin^2 a x\,dx = \frac{x}{2} - \frac{\sin 2 a x}{4 a}$$
73. $$\int \sin^3 a x\,dx = - \frac{3 \cos a x}{4 a} + \frac{\cos 3 a x}{12 a}$$
74. $$\int \sin^n a x\,dx = - \frac{1}{a} \cos a x \, {}_2F_1 \left[ \frac{1}{2} , \frac{1 - n}{2} , \frac{3}{2} , \cos^2 a x \right]$$
75. $$\int \cos a x\,dx = \frac{1}{a} \sin a x$$
76. $$\int \cos^2 a x\,dx = \frac{x}{2} + \frac{\sin 2 a x}{4 a}$$
77. $$\int \cos^3 a x\,dx = \frac{3 \sin a x}{4 a} + \frac{\sin 3 a x}{12 a}$$
78. $$\int \cos^p a x\,dx = - \frac{1}{a ( 1 + p )} \cos^{1 + p} a x \times {}_2F_1 \left[ \frac{1 + p}{2} , \frac{1}{2} , \frac{3 + p}{2} , \cos^2 a x \right]$$
79. $$\int \cos x \sin x\,dx = \frac{1}{2} \sin^2 x + {c}_1 = - \frac{1}{2} \cos^2 x + {c}_2 = - \frac{1}{4} \cos 2 x + {c}_3$$
80. $$\int \cos a x \sin b x\,dx = \frac{\cos [ ( a - b ) x ]}{2 ( a - b )} - \frac{\cos [ ( a + b ) x ]}{2 ( a + b )} , a \neq b$$
81. $$\int \sin^2 a x \cos b x\,dx = - \frac{\sin [ ( 2 a - b ) x ]}{4 ( 2 a - b )} + \frac{\sin b x}{2 b} - \frac{\sin [ ( 2 a + b ) x ]}{4 ( 2 a + b )}$$
82. $$\int \sin^2 x \cos x\,dx = \frac{1}{3} \sin^3 x$$
83. $$\int \cos^2 a x \sin b x\,dx = \frac{\cos [ ( 2 a - b ) x ]}{4 ( 2 a - b )} - \frac{\cos b x}{2 b} - \frac{\cos [ ( 2 a + b ) x ]}{4 ( 2 a + b )}$$
84. $$\int \cos^2 a x \sin a x\,dx = - \frac{1}{3 a} \cos^3 a x$$
85. $$\int \sin^2 a x \cos^2 b x\,dx = \frac{x}{4} - \frac{\sin 2 a x}{8 a} - \frac{\sin [ 2 ( a - b ) x ]}{16 ( a - b )} + \frac{\sin 2 b x}{8 b} - \frac{\sin [ 2 ( a + b ) x ]}{16 ( a + b )}$$
86. $$\int \sin^2 a x \cos^2 a x\,dx = \frac{x}{8} - \frac{\sin 4 a x}{32 a}$$
87. $$\int \tan a x\,dx = - \frac{1}{a} \ln \cos a x$$
88. $$\int \tan^2 a x\,dx = - x + \frac{1}{a} \tan a x$$
89. $$\int \tan^n a x\,dx = \frac{\tan^{n + 1} a x}{a ( 1 + n )} \times {}_2F_1 \left( \frac{n + 1}{2} , 1 , \frac{n + 3}{2} , - \tan^2 a x \right)$$
90. $$\int \tan^3 a x\,dx = \frac{1}{a} \ln \cos a x + \frac{1}{2 a} \sec^2 a x$$
91. $$\int \sec x\,dx = \ln | \sec x + \tan x | = 2 \tanh^{- 1} \left( \tan \frac{x}{2} \right)$$
92. $$\int \sec^2 a x\,dx = \frac{1}{a} \tan a x$$
93. $$\int \sec^3 x\,dx = \frac{1}{2} \sec x \tan x + \frac{1}{2} \ln | \sec x + \tan x |$$
94. $$\int \sec x \tan x\,dx = \sec x$$
95. $$\int \sec^2 x \tan x\,dx = \frac{1}{2} \sec^2 x$$
96. $$\int \sec^n x \tan x\,dx = \frac{1}{n} \sec^n x , n \neq 0$$
97. $$\int \csc x\,dx = \ln \left| \tan \frac{x}{2} \right| = \ln | \csc x - \cot x | + C$$
98. $$\int \csc^2 a x\,dx = - \frac{1}{a} \cot a x$$
99. $$\int \csc^3 x\,dx = - \frac{1}{2} \cot x \csc x + \frac{1}{2} \ln | \csc x - \cot x |$$
100. $$\int \csc^n x \cot x\,dx = - \frac{1}{n} \csc^n x , n \neq 0$$
101. $$\int \sec x \csc x\,dx = \ln | \tan x |$$

انتگرال چندجمله‌ای‌های مثلثاتی

102. $$\int x \cos x\,dx = \cos x + x \sin x$$
103. $$\int x \cos a x\,dx = \frac{1}{a^2} \cos a x + \frac{x}{a} \sin a x$$
104. $$\int x^2 \cos x\,dx = 2 x \cos x + \left( x^2 - 2 \right) \sin x$$
105. $$\int x^2 \cos a x\,dx = \frac{2 x \cos a x}{a^2} + \frac{a^2 x^2 - 2}{a^3} \sin a x$$
106. $$\int x^n \cos x\,dx = - \frac{1}{2} {( i )}^{n + 1} \left[ \Gamma ( n + 1 , - i x ) + {( - 1 )}^n \Gamma ( n + 1 , i x ) \right]$$
107. $$\int x^n \cos a x\,dx = \frac{1}{2} {( i a )}^{1 - n} \left[ {( - 1 )}^n \Gamma ( n + 1 , - i a x ) - \Gamma ( n + 1 , i x a ) \right]$$
108. $$\int x \sin x\,dx = - x \cos x + \sin x$$
109. $$\int x \sin a x\,dx = - \frac{x \cos a x}{a} + \frac{\sin a x}{a^2}$$
110. $$\int x^2 \sin x\,dx = \left( 2 - x^2 \right) \cos x + 2 x \sin x$$
111. $$\int x^2 \sin a x\,dx = \frac{2 - a^2 x^2}{a^3} \cos a x + \frac{2 x \sin a x}{a^2}$$
112. $$\int x^n \sin x\,dx = - \frac{1}{2} {( i )}^n \left[ \Gamma ( n + 1 , - i x ) - {( - 1 )}^n \Gamma ( n + 1 , i x ) \right]$$
113. $$\int x \cos^2 x\,dx = \frac{x^2}{4} + \frac{1}{8} \cos 2 x + \frac{1}{4} x \sin 2 x$$
114. $$\int x \sin^2 x\,dx = \frac{x^2}{4} - \frac{1}{8} \cos 2 x - \frac{1}{4} x \sin 2 x$$
115. $$\int x \tan^2 x\,dx = - \frac{x^2}{2} + \ln \cos x + x \tan x$$
116. $$\int x \sec^2 x\,dx = \ln \cos x + x \tan x$$

انتگرال‌های حاصل‌ضرب مثلثات و توابع توانی

117. $$\int e^x \sin x\,dx = \frac{1}{2} e^x ( \sin x - \cos x )$$
118. $$\int e^{b x} \sin a x\,dx = \frac{1}{a^2 + b^2} e^{b x} ( b \sin a x - a \cos a x )$$
119. $$\int e^x \cos x\,dx = \frac{1}{2} e^x ( \sin x + \cos x )$$
120. $$\int e^{b x} \cos a x\,dx = \frac{1}{a^2 + b^2} e^{b x} ( a \sin a x + b \cos a x )$$
121. $$\int x e^x \sin x\,dx = \frac{1}{2} e^x ( \cos x - x \cos x + x \sin x )$$
122. $$\int x e^x \cos x\,dx = \frac{1}{2} e^x ( x \cos x - \sin x + x \sin x )$$

انتگرال توابع هیپربولیک

123. $$\int \cosh a x\,dx = \frac{1}{a} \sinh a x$$
124. $$\int e^{a x} \cosh b x\,dx = \left\{ \begin{array}{ll} \displaystyle \frac{e^{a x}}{a^2 - b^2} [ a \cosh b x - b \sinh b x ] \, & \displaystyle a \neq b \\ \displaystyle \frac{e^{2 a x}}{4 a} + \frac{x}{2} \, & \displaystyle a = b \end{array} \right.$$
125. $$\int \sinh a x\,dx = \frac{1}{a} \cosh a x$$
126. $$\int e^{a x} \sinh b x\,dx = \left\{ \begin{array}{ll} \displaystyle \frac{e^{a x}}{a^2 - b^2} [ - b \cosh b x + a \sinh b x ] \, & \displaystyle a \neq b \\ \displaystyle \frac{e^{2 a x}}{4 a} - \frac{x}{2} \, & \displaystyle a = b \end{array} \right.$$
127. $$\int \tanh a x\,dx = \frac{1}{a} \ln \cosh a x$$
128. $$\int e^{a x} \tanh b x\,dx = \left\{ \begin{array}{ll} \begin{aligned} & \dfrac{e^{(a + 2 b) x}}{a + 2 b}\, {}_2F_1\!\left[ 1 + \dfrac{a}{2 b}, 1, 2 + \dfrac{a}{2 b}, -e^{2 b x} \right] \\ & - \dfrac{1}{a} e^{a x}\, {}_2F_1\!\left[ 1, \dfrac{a}{2 b}, 1 + \dfrac{a}{2 b}, -e^{2 b x} \right], \end{aligned} & a \neq b \\[2ex] \dfrac{e^{a x} - 2 \tan^{-1}\left[ e^{a x} \right]}{a}, & a = b \end{array} \right.$$
129. $$\int \cos a x \cosh b x\,dx = \frac{1}{a^2 + b^2} \left[ a \sin a x \cosh b x + b \cos a x \sinh b x \right]$$
130. $$\int \cos a x \sinh b x\,dx = \frac{1}{a^2 + b^2} \left[ b \cos a x \cosh b x + a \sin a x \sinh b x \right]$$
131. $$\int \sin a x \cosh b x\,dx = \frac{1}{a^2 + b^2} \left[ - a \cos a x \cosh b x + b \sin a x \sinh b x \right]$$
132. $$\int \sin a x \sinh b x\,dx = \frac{1}{a^2 + b^2} \left[ b \cosh b x \sin a x - a \cos a x \sinh b x \right]$$
133. $$\int \sinh a x \cosh a x\,dx = \frac{1}{4 a} \left[ - 2 a x + \sinh 2 a x \right]$$
134. $$\int \sinh a x \cosh b x\,dx = \frac{1}{b^2 - a^2} \left[ b \cosh b x \sinh a x - a \cosh a x \sinh b x \right]$$

تهیه و گردآوری: گروه آموزشی رایشمند - مرجع تخصصی ریاضیات