update 专升本 高等数学及其应用 一元函数积分学及其应用 积分表

Author: Yue-plus

docs/20-专升本/高等数学及其应用/99-积分表.md

@@ -2,23 +2,23 @@
 
 ## (一) 含有 $ax+b$ 的积分
 
-01. $\int \frac{\mathrm{d} x}{ax+b} = \frac{1}{a} \ln |ax+b| + C$
+1. $\int \frac{\mathrm{d} x}{ax+b} = \frac{1}{a} \ln |ax+b| + C$
 
-02. $\int (ax+b)^{\mu} \mathrm{d} x = \frac{1}{a(\mu+1)} (ax+b)^{\mu+1} + C \quad (\mu \neq -1)$
+2. $\int (ax+b)^{\mu} \mathrm{d} x = \frac{1}{a(\mu+1)} (ax+b)^{\mu+1} + C \quad (\mu \neq -1)$
 
-03. $\int \frac{x}{ax+b} \mathrm{d} x = \frac{1}{a^2} \left(ax + b - b \ln |ax+b| \right) + C$
+3. $\int \frac{x}{ax+b} \mathrm{d} x = \frac{1}{a^2} \left(ax + b - b \ln |ax+b| \right) + C$
 
-04. $\int \frac{x^2}{ax+b} \mathrm{d} x = \frac{1}{a^3} \left[ \frac{1}{2} (ax+b)^2 - 2b (ax+b) + b^2 \ln |ax+b| \right] + C$
+4. $\int \frac{x^2}{ax+b} \mathrm{d} x = \frac{1}{a^3} \left[ \frac{1}{2} (ax+b)^2 - 2b (ax+b) + b^2 \ln |ax+b| \right] + C$
 
-05. $\int \frac{\mathrm{d} x}{x(ax+b)} = -\frac{1}{b} \ln \left| \frac{ax+b}{x} \right| + C$
+5. $\int \frac{\mathrm{d} x}{x(ax+b)} = -\frac{1}{b} \ln \left| \frac{ax+b}{x} \right| + C$
 
-06. $\int \frac{\mathrm{d} x}{x^2(ax+b)} = -\frac{1}{bx} + \frac{a}{b^2} \ln \left| \frac{ax+b}{x} \right| + C$
+6. $\int \frac{\mathrm{d} x}{x^2(ax+b)} = -\frac{1}{bx} + \frac{a}{b^2} \ln \left| \frac{ax+b}{x} \right| + C$
 
-07. $\int \frac{x}{(ax+b)^2} \mathrm{d} x = \frac{1}{a^2} \left( \ln |ax+b| + \frac{b}{ax+b} \right) + C$
+7. $\int \frac{x}{(ax+b)^2} \mathrm{d} x = \frac{1}{a^2} \left( \ln |ax+b| + \frac{b}{ax+b} \right) + C$
 
-08. $\int \frac{x^2}{(ax+b)^2} \mathrm{d} x = \frac{1}{a^3} \left(ax + b - 2b \ln |ax+b| - \frac{b^2}{ax+b} \right) + C$
+8. $\int \frac{x^2}{(ax+b)^2} \mathrm{d} x = \frac{1}{a^3} \left(ax + b - 2b \ln |ax+b| - \frac{b^2}{ax+b} \right) + C$
 
-09. $\int \frac{\mathrm{d} x}{x(ax+b)^2} = \frac{1}{b(ax+b)} - \frac{1}{b^2} \ln \left| \frac{ax+b}{x} \right| + C$
+9. $\int \frac{\mathrm{d} x}{x(ax+b)^2} = \frac{1}{b(ax+b)} - \frac{1}{b^2} \ln \left| \frac{ax+b}{x} \right| + C$
 
 ## (二) 含有 $\sqrt{ax+b}$ 的积分
 
@@ -42,3 +42,313 @@
 17. $\int \frac{\sqrt{ax+b}}{x} \, \mathrm{d} x = 2 \sqrt{ax+b} + b \int \frac{\mathrm{d} x}{x \sqrt{ax+b}}$
 
 18. $\int \frac{\sqrt{ax+b}}{x^2} \, \mathrm{d} x = -\frac{\sqrt{ax+b}}{x} + \frac{a}{2} \int \frac{\mathrm{d} x}{x \sqrt{ax+b}}$
+
+## (三) 含有 $x^2 \pm a^2$ 的积分
+
+19. $\int \frac{\mathrm{d} x}{x^2 + a^2} = \frac{1}{a} \arctan \frac{x}{a} + C$
+
+20. $\int \frac{\mathrm{d} x}{(x^2 + a^2)^n} = \frac{x}{2(n-1)a^2(x^2 + a^2)^{n-1}} + \frac{2n-3}{2(n-1)a^2} \int \frac{\mathrm{d} x}{(x^2 + a^2)^{n-1}}$
+
+21. $\int \frac{\mathrm{d} x}{x^2 - a^2} = \frac{1}{2a} \ln \left| \frac{x-a}{x+a} \right| + C$
+
+## (四) 含有 $ax^2 + b \quad (a > 0)$ 的积分
+
+22. $\int \frac{\mathrm{d} x}{ax^2 + b} = \begin{cases} 
+        \frac{1}{\sqrt{ab}} \arctan \sqrt{\frac{a}{b}} x + C & (b > 0) \\
+        \frac{1}{2 \sqrt{-ab}} \ln \left| \frac{\sqrt{a} x - \sqrt{-b}}{\sqrt{a} x + \sqrt{-b}} \right| + C & (b < 0)
+    \end{cases}$
+
+23. $\int \frac{x}{ax^2 + b} \mathrm{d} x = \frac{1}{2a} \ln \left| ax^2 + b \right| + C$
+
+24. $\int \frac{x^2}{ax^2 + b} \mathrm{d} x = \frac{x}{a} - \frac{b}{a} \int \frac{\mathrm{d} x}{ax^2 + b}$
+
+25. $\int \frac{\mathrm{d} x}{x(ax^2 + b)} = \frac{1}{2b} \ln \frac{x^2}{| ax^2 + b |} + C$
+
+26. $\int \frac{\mathrm{d} x}{x^2 (ax^2 + b)} = -\frac{1}{bx} - \frac{a}{b} \int \frac{\mathrm{d} x}{ax^2 + b}$
+
+27. $\int \frac{\mathrm{d} x}{x^3 (ax^2 + b)} = \frac{a}{2b^2} \ln \frac{| ax^2 + b |}{x^2} - \frac{1}{2bx^2} + C$
+
+28. $\int \frac{\mathrm{d} x}{(ax^2 + b)^2} = \frac{x}{2b(ax^2 + b)} + \frac{1}{2b} \int \frac{\mathrm{d} x}{ax^2 + b}$
+
+## (五) 含有 $ax^2 + bx + c \quad (a > 0)$ 的积分
+
+29. $\int \frac{\mathrm{d} x}{ax^2 + bx + c} = \begin{cases} 
+        \frac{2}{\sqrt{4ac - b^2}} \arctan \left( \frac{2ax + b}{\sqrt{4ac - b^2}} \right) + C & (b^2 < 4ac) \\
+        \frac{1}{\sqrt{b^2 - 4ac}} \ln \left| \frac{2ax + b - \sqrt{b^2 - 4ac}}{2ax + b + \sqrt{b^2 - 4ac}} \right| + C & (b^2 > 4ac)
+    \end{cases}$
+
+30. $\int \frac{x}{ax^2 + bx + c} \mathrm{d} x = \frac{1}{2a} \ln \left| ax^2 + bx + c \right| - \frac{b}{2a} \int \frac{\mathrm{d} x}{ax^2 + bx + c}$
+
+## (六) 含有 $\sqrt{x^2+a^2} \quad (a>0)$ 的积分
+
+31. $\int \frac{\mathrm{d} x}{\sqrt{x^2+a^2}} = \text{arsh} \frac{x}{a} + C_1 = \ln(x + \sqrt{x^2 + a^2}) + C$
+
+32. $\int \frac{\mathrm{d} x}{\sqrt{(x^2+a^2)^3}} = \frac{x}{a^2 \sqrt{x^2 + a^2}} + C$
+
+33. $\int \frac{x}{\sqrt{x^2 + a^2}} \, \mathrm{d} x = \sqrt{x^2 + a^2} + C$
+
+34. $\int \frac{x}{\sqrt{(x^2 + a^2)^3}} \mathrm{d} x = -\frac{1}{\sqrt{x^2 + a^2}} + C$
+
+35. $\int \frac{x^2}{\sqrt{x^2 + a^2}} \, \mathrm{d} x = \frac{x}{2} \sqrt{x^2 + a^2} - \frac{a^2}{2} \ln(x + \sqrt{x^2 + a^2}) + C$
+
+36. $\int \frac{x^2}{\sqrt{(x^2 + a^2)^3}} \, \mathrm{d} x = -\frac{x}{\sqrt{x^2 + a^2}} + \ln(x + \sqrt{x^2 + a^2}) + C$
+
+37. $\int \frac{\mathrm{d} x}{x \sqrt{x^2 + a^2}} = \frac{1}{a} \ln \frac{\sqrt{x^2 + a^2} - a}{|x|} + C$
+
+38. $\int \frac{\mathrm{d} x}{x^2 \sqrt{x^2 + a^2}} = -\frac{\sqrt{x^2 + a^2}}{a^2 x} + C$
+
+39. $\int \sqrt{x^2 + a^2} \, \mathrm{d} x = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln(x + \sqrt{x^2 + a^2}) + C$
+
+40. $\int \sqrt{(x^2 + a^2)^3} \, \mathrm{d} x = \frac{x}{8} (2x^2 + 5a^2) \sqrt{x^2 + a^2} + \frac{3}{8} a^4 \ln(x + \sqrt{x^2 + a^2}) + C$
+
+41. $\int x \sqrt{x^2 + a^2} \, \mathrm{d} x = \frac{1}{3} \sqrt{(x^2 + a^2)^3} + C$
+
+42. $\int x^2 \sqrt{x^2 + a^2} \, \mathrm{d} x = \frac{x}{8} (2x^2 + a^2) \sqrt{x^2 + a^2} - \frac{a^4}{8} \ln(x + \sqrt{x^2 + a^2}) + C$
+
+43. $\int \frac{\sqrt{x^2 + a^2}}{x} \, \mathrm{d} x = \sqrt{x^2 + a^2} + a \ln \frac{\sqrt{x^2 + a^2} - a}{|x|} + C$
+
+44. $\int \frac{\sqrt{x^2 + a^2}}{x^2} \, \mathrm{d} x = -\frac{\sqrt{x^2 + a^2}}{x} + \ln(x + \sqrt{x^2 + a^2}) + C$
+
+## (七) 含有 $\sqrt{x^2 - a^2} \quad (a > 0)$ 的积分
+
+45. $\int \frac{\mathrm{d} x}{\sqrt{x^2 - a^2}} = \frac{x}{|x|} \text{arch} \frac{|x|}{a} + C_1 = \ln \left| x + \sqrt{x^2 - a^2} \right| + C$
+
+46. $\int \frac{\mathrm{d} x}{\left( \sqrt{x^2 - a^2} \right)^3} = - \frac{x}{a^2 \sqrt{x^2 - a^2}} + C$
+
+47. $\int \frac{x}{\sqrt{x^2 - a^2}} \mathrm{d} x = \sqrt{x^2 - a^2} + C$
+
+48. $\int \frac{x}{\sqrt{(x^2 - a^2)^3}} \mathrm{d} x = - \frac{1}{\sqrt{x^2 - a^2}} + C$
+
+49. $\int \frac{x^2}{\sqrt{x^2 - a^2}} \mathrm{d} x = \frac{x}{2} \sqrt{x^2 - a^2} + \frac{a^2}{2} \ln \left| x + \sqrt{x^2 - a^2} \right| + C$
+
+50. $\int \frac{x^2}{\left( \sqrt{x^2 - a^2} \right)^3} \mathrm{d} x = - \frac{x}{\sqrt{x^2 - a^2}} + \ln \left| x + \sqrt{x^2 - a^2} \right| + C$
+
+51. $\int \frac{\mathrm{d} x}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \text{arccos} \frac{a}{|x|} + C$
+
+52. $\int \frac{\mathrm{d} x}{x^2 \sqrt{x^2 - a^2}} = \frac{\sqrt{x^2 - a^2}}{a^2 x} + C$
+
+53. $\int \sqrt{x^2 - a^2} \mathrm{d} x = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \ln \left| x + \sqrt{x^2 - a^2} \right| + C$
+
+54. $\int \sqrt{(x^2 - a^2)^3} \, \mathrm{d} x = \frac{x}{8} \left(2x^2 - 5a^2\right) \sqrt{x^2 - a^2} + \frac{3}{8} a^4 \ln \left| x + \sqrt{x^2 - a^2} \right| + C$
+
+55. $\int x \sqrt{x^2 - a^2} \, \mathrm{d} x = \frac{1}{3} \sqrt{(x^2 - a^2)^3} + C$
+
+56. $\int x^2 \sqrt{x^2 - a^2} \, \mathrm{d} x = \frac{x}{8} \left(2x^2 - a^2\right) \sqrt{x^2 - a^2} - \frac{a^4}{8} \ln \left| x + \sqrt{x^2 - a^2} \right| + C$
+
+57. $\int \frac{\sqrt{x^2 - a^2}}{x} \, \mathrm{d} x = \sqrt{x^2 - a^2} - a \arccos \frac{a}{|x|} + C$
+
+58. $\int \frac{\sqrt{x^2 - a^2}}{x^2} \, \mathrm{d} x = -\frac{\sqrt{x^2 - a^2}}{x} + \ln \left| x + \sqrt{x^2 - a^2} \right| + C$
+
+## (八) 含有 $\sqrt{a^2 - x^2} \quad (a > 0)$ 的积分
+
+59. $\int \frac{\mathrm{d} x}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C$
+
+60. $\int \frac{\mathrm{d} x}{\left( \sqrt{a^2 - x^2} \right)^3} = \frac{x}{a^2 \sqrt{a^2 - x^2}} + C$
+
+61. $\int \frac{x}{\sqrt{a^2 - x^2}} \mathrm{d} x = -\sqrt{a^2 - x^2} + C$
+
+62. $\int \frac{x}{\sqrt{(a^2 - x^2)^3}} \mathrm{d} x = \frac{1}{\sqrt{a^2 - x^2}} + C$
+
+63. $\int \frac{x^2}{\sqrt{a^2 - x^2}} \mathrm{d} x = -\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \arcsin \frac{x}{a} + C$
+
+64. $\int \frac{x^2}{\sqrt{(a^2 - x^2)^3}} \mathrm{d} x = \frac{x}{\sqrt{a^2 - x^2}} - \arcsin \frac{x}{a} + C$
+
+65. $\int \frac{\mathrm{d} x}{x \sqrt{a^2 - x^2}} = \frac{1}{a} \ln \frac{a - \sqrt{a^2 - x^2}}{|x|} + C$
+
+66. $\int \frac{\mathrm{d} x}{x^2 \sqrt{a^2 - x^2}} = -\frac{\sqrt{a^2 - x^2}}{a^2 x} + C$
+
+67. $\int \sqrt{a^2 - x^2} \mathrm{d} x = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \arcsin \frac{x}{a} + C$
+
+68. $\int \sqrt{\left( a^2 - x^2 \right)^3} \mathrm{d} x = \frac{x}{8} \left( 5a^2 - 2x^2 \right) \sqrt{a^2 - x^2} + \frac{3}{8} a^4 \arcsin \frac{x}{a} + C$
+
+69. $\int x \sqrt{a^2 - x^2} \mathrm{d} x = -\frac{1}{3} \sqrt{\left( a^2 - x^2 \right)^3} + C$
+
+70. $\int x^2 \sqrt{a^2 - x^2} \mathrm{d} x = \frac{x}{8} (2x^2 - a^2) \sqrt{a^2 - x^2} + \frac{a^4}{8} \arcsin \frac{x}{a} + C$
+
+71. $\int \frac{\sqrt{a^2 - x^2}}{x} \mathrm{d} x = \sqrt{a^2 - x^2} + a \ln \frac{a - \sqrt{a^2 - x^2}}{|x|} + C$
+
+72. $\int \frac{\sqrt{a^2 - x^2}}{x^2} \mathrm{d} x = - \frac{\sqrt{a^2 - x^2}}{x} - \arcsin \frac{x}{a} + C$
+
+## (九) 含有 $\sqrt{\pm ax^2 + bx + c} \quad (a > 0)$ 的积分
+
+73. $\int \frac{\mathrm{d} x}{\sqrt{ax^2 + bx + c}} = \frac{1}{\sqrt{a}} \ln \left| 2ax + b + 2 \sqrt{a} \sqrt{ax^2 + bx + c} \right| + C$
+
+74. $\int \sqrt{ax^2 + bx + c} \mathrm{d} x = \frac{2ax + b}{4a} \sqrt{ax^2 + bx + c} + \frac{4ac - b^2}{8 \sqrt{a^3}} \ln \left| 2ax + b + 2 \sqrt{a} \sqrt{ax^2 + bx + c} \right| + C$
+
+75. $\int \frac{x}{\sqrt{ax^2 + bx + c}} \mathrm{d} x = \frac{1}{a} \sqrt{ax^2 + bx + c} - \frac{b}{2 \sqrt{a^3}} \ln \left| 2ax + b + 2 \sqrt{a} \sqrt{ax^2 + bx + c} \right| + C$
+
+76. $\int \frac{\mathrm{d} x}{\sqrt{c + bx - ax^2}} = \frac{1}{\sqrt{a}} \arcsin \frac{2ax - b}{\sqrt{b^2 + 4ac}} + C$
+
+77. $\int \sqrt{c + bx - ax^2} \mathrm{d} x = \frac{2ax - b}{4a} \sqrt{c + bx - ax^2} + \frac{b^2 + 4ac}{8 \sqrt{a^3}} \arcsin \frac{2ax - b}{\sqrt{b^2 + 4ac}} + C$
+
+78. $\int \frac{x}{\sqrt{c + bx - ax^2}} \mathrm{d} x = - \frac{1}{a} \sqrt{c + bx - ax^2} + \frac{b}{2 \sqrt{a^3}} \arcsin \frac{2ax - b}{\sqrt{b^2 + 4ac}} + C$
+
+## (十) 含有 $\sqrt{\pm \frac{x-a}{x-b}}$ 或 $\sqrt{(x-a)(b-x)}$ 的积分
+
+79. $\int \sqrt{\frac{x-a}{x-b}} \, \mathrm{d} x = (x-b) \sqrt{\frac{x-a}{x-b}} + (b-a) \ln \left( \sqrt{|x-a|} + \sqrt{|x-b|} \right) + C$
+
+80. $\int \sqrt{\frac{x-a}{b-x}} \, \mathrm{d} x = (x-b) \sqrt{\frac{x-a}{b-x}} + (b-a) \arcsin \sqrt{\frac{x-a}{b-a}} + C$
+
+81. $\int \frac{\mathrm{d} x}{\sqrt{(x-a)(b-x)}} = 2 \arcsin \sqrt{\frac{x-a}{b-a}} + C \quad (a < b)$
+
+82. $\int \sqrt{(x-a)(b-x)} \, \mathrm{d} x = \frac{2x-a-b}{4} \sqrt{(x-a)(b-x)} + \frac{(b-a)^2}{4} \arcsin \sqrt{\frac{x-a}{b-a}} + C \quad (a < b)$
+
+## (十一) 含有三角函数的积分
+
+83. $\int \sin x \, \mathrm{d} x = -\cos x + C$
+
+84. $\int \cos x \, \mathrm{d} x = \sin x + C$
+
+85. $\int \tan x \, \mathrm{d} x = -\ln | \cos x | + C$
+
+86. $\int \cot x \, \mathrm{d} x = \ln | \sin x | + C$
+
+87. $\int \sec x \, \mathrm{d} x = \ln \left| \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) \right| + C = \ln | \sec x + \tan x | + C$
+
+88. $\int \csc x \, \mathrm{d} x = \ln \left| \tan \frac{x}{2} \right| + C = \ln | \csc x - \cot x | + C$
+
+89. $\int \sec^2 x \, \mathrm{d} x = \tan x + C$
+
+90. $\int \csc^2 x \, \mathrm{d} x = -\cot x + C$
+
+91. $\int \sec x \tan x \, \mathrm{d} x = \sec x + C$
+
+92. $\int \csc x \cot x \, \mathrm{d} x = -\csc x + C$
+
+93. $\int \sin^2 x \, \mathrm{d} x = \frac{x}{2} - \frac{1}{4} \sin 2x + C$
+
+94. $\int \cos^2 x \, \mathrm{d} x = \frac{x}{2} + \frac{1}{4} \sin 2x + C$
+
+95. $\int \sin^n x \, \mathrm{d} x = \frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, \mathrm{d} x$
+
+96. $\int \cos^n x \, \mathrm{d} x = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x \, \mathrm{d} x$
+
+97. $\int \frac{\mathrm{d} x}{\sin^n x} = -\frac{1}{n-1} \cdot \frac{\cos x}{\sin^{n-1} x} + \frac{n-2}{n-1} \int \frac{\mathrm{d} x}{\sin^{n-2} x}$
+
+98. $\int \frac{\mathrm{d} x}{\cos^n x} = \frac{1}{n-1} \cdot \frac{\sin x}{\cos^{n-1} x} + \frac{n-2}{n-1} \int \frac{\mathrm{d} x}{\cos^{n-2} x}$
+
+99. $\int \cos^m x \sin^n x \, \mathrm{d} x \\
+    = \frac{1}{m+n} \cos^{m-1} x \sin^{n+1} x + \frac{m-1}{m+n} \int \cos^{m-2} x \sin^n x \, \mathrm{d} x \\
+    = -\frac{1}{m+n} \cos^{m+1} x \sin^{n-1} x + \frac{n-1}{m+n} \int \cos^m x \sin^{n-2} x \, \mathrm{d} x$
+
+100. $\int \sin ax \cos bx \, \mathrm{d} x = -\frac{1}{2(a+b)} \cos (a+b)x - \frac{1}{2(a-b)} \cos (a-b)x + C$
+
+101. $\int \sin ax \sin bx \, \mathrm{d} x = -\frac{1}{2(a+b)} \sin (a+b)x + \frac{1}{2(a-b)} \sin (a-b)x + C$
+
+102. $\int \cos ax \cos bx\mathrm{d} x = \frac{1}{2(a+b)}\sin (a+b)x + \frac{1}{2(a-b)}\sin (a-b)x + C$
+
+103. $\int \frac{\mathrm{d} x}{a+b\sin x} = \frac{2}{\sqrt{a^2-b^2}} \arctan \frac{a \tan \frac{x}{2} + b}{\sqrt{a^2-b^2}} + C \quad (a^2 > b^2)$
+
+104. $\int \frac{\mathrm{d} x}{a+b\sin x} = \frac{1}{\sqrt{b^2-a^2}} \ln \left| \frac{a \tan \frac{x}{2} + b - \sqrt{b^2-a^2}}{a \tan \frac{x}{2} + b + \sqrt{b^2-a^2}} \right| + C \quad (a^2 < b^2)$
+
+105. $\int \frac{\mathrm{d} x}{a+b\cos x} = \frac{2}{a+b} \sqrt{\frac{a+b}{a-b}} \arctan \left( \sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2} \right) + C \quad (a^2 > b^2)$
+
+106. $\int \frac{\mathrm{d} x}{a+b\cos x} = \frac{1}{a+b} \sqrt{\frac{a+b}{b-a}} \ln \left| \frac{\tan \frac{x}{2} + \sqrt{\frac{a+b}{b-a}}}{\tan \frac{x}{2} - \sqrt{\frac{a+b}{b-a}}} \right| + C \quad (a^2 < b^2)$
+
+107. $\int \frac{\mathrm{d} x}{a^2 \cos^2 x + b^2 \sin^2 x} = \frac{1}{ab} \arctan \left( \frac{b}{a} \tan x \right) + C$
+
+108. $\int \frac{\mathrm{d} x}{a^2 \cos^2 x - b^2 \sin^2 x} = \frac{1}{2ab} \ln \left| \frac{b \tan x + a}{b \tan x - a} \right| + C$
+
+109. $\int x \sin ax \mathrm{d} x = \frac{1}{a^2} \sin ax - \frac{1}{a} x \cos ax + C$
+
+110. $\int x^2 \sin ax \mathrm{d} x = -\frac{1}{a} x^2 \cos ax + \frac{2}{a^2} x \sin ax + \frac{2}{a^3} \cos ax + C$
+
+111. $\int x \cos ax \mathrm{d} x = \frac{1}{a^2} \cos ax + \frac{1}{a} x \sin ax + C$
+
+112. $\int x^2 \cos ax \mathrm{d} x = \frac{1}{a} x^2 \sin ax + \frac{2}{a^2} x \cos ax - \frac{2}{a^3} \sin ax + C$
+
+## (十二) 含有反三角函数的积分 (其中 $a > 0$)
+
+113. $\int \arcsin \frac{x}{a} \mathrm{d} x = x \arcsin \frac{x}{a} + \sqrt{a^2 - x^2} + C$
+
+114. $\int x \arcsin \frac{x}{a} \mathrm{d} x = \left( \frac{x^2}{2} - \frac{a^2}{4} \right) \arcsin \frac{x}{a} + \frac{x}{4} \sqrt{a^2 - x^2} + C$
+
+115. $\int x^2 \arcsin \frac{x}{a} \mathrm{d} x = \frac{x^3}{3} \arcsin \frac{x}{a} + \frac{1}{9} (x^2 + 2a^2) \sqrt{a^2 - x^2} + C$
+
+116. $\int \arccos \frac{x}{a} \mathrm{d} x = x \arccos \frac{x}{a} - \sqrt{a^2 - x^2} + C$
+
+117. $\int x \arccos \frac{x}{a} \,\mathrm{d} x = \left( \frac{x^2}{2} - \frac{a^2}{4} \right) \arccos \frac{x}{a} - \frac{x}{4} \sqrt{a^2 - x^2} + C$
+
+118. $\int x^2 \arccos \frac{x}{a} \,\mathrm{d} x = \frac{x^3}{3} \arccos \frac{x}{a} - \frac{1}{9} (x^2 + 2a^2) \sqrt{a^2 - x^2} + C$
+
+119. $\int \arctan \frac{x}{a} \,\mathrm{d} x = x \arctan \frac{x}{a} - \frac{a}{2} \ln \left( a^2 + x^2 \right) + C$
+
+120. $\int x \arctan \frac{x}{a} \,\mathrm{d} x = \frac{1}{2} \left( a^2 + x^2 \right) \arctan \frac{x}{a} - \frac{a}{2} x + C$
+
+121. $\int x^2 \arctan \frac{x}{a} \,\mathrm{d} x = \frac{x^3}{3} \arctan \frac{x}{a} - \frac{a}{6} x^2 + \frac{a^3}{6} \ln \left( a^2 + x^2 \right) + C$
+
+## (十三) 含有指数函数的积分
+
+122. $\int a^x \,\mathrm{d} x = \frac{1}{\ln a} a^x + C$
+
+123. $\int e^{ax} \,\mathrm{d} x = \frac{1}{a} e^{ax} + C$
+
+124. $\int x e^{ax} \,\mathrm{d} x = \frac{1}{a^2} (ax - 1) e^{ax} + C$
+
+125. $\int x^n e^{ax} \,\mathrm{d} x = \frac{1}{a} x^n e^{ax} - \frac{n}{a} \int x^{n-1} e^{ax} \,\mathrm{d} x$
+
+126. $\int x a^x \,\mathrm{d} x = \frac{x}{\ln a} a^x - \frac{1}{(\ln a)^2} a^x + C$
+
+127. $\int x^n a^x \,\mathrm{d} x = \frac{1}{\ln a} x^n a^x - \frac{n}{\ln a} \int x^{n-1} a^x \,\mathrm{d} x$
+
+128. $\int e^{ax} \sin bx \,\mathrm{d} x = \frac{1}{a^2 + b^2} e^{ax} (a \sin bx - b \cos bx) + C$
+
+129. $\int e^{ax} \cos bx \,\mathrm{d} x = \frac{1}{a^2 + b^2} e^{ax} (b \sin bx + a \cos bx) + C$
+
+130. $\int e^{ax} \sin^n bx \,\mathrm{d} x = \frac{1}{a^2 + b^2 n^2} e^{ax} \sin^{n-1} bx \left( a \sin bx - n b \cos bx \right) + \frac{n (n-1) b^2}{a^2 + b^2 n^2} \int e^{ax} \sin^{n-2} bx \,\mathrm{d} x$
+
+131. $\int e^{ax} \cos^n bx \,\mathrm{d} x = \frac{1}{a^2 + b^2 n^2} e^{ax} \cos^{n-1} bx \left( a \cos bx + n b \sin bx \right) + \frac{n (n-1) b^2}{a^2 + b^2 n^2} \int e^{ax} \cos^{n-2} bx \,\mathrm{d} x$
+
+## (十四) 含有对数函数的积分
+
+132. $\int \ln x \,\mathrm{d} x = x \ln x - x + C$
+
+133. $\int \frac{\mathrm{d} x}{x \ln x} = \ln |\ln x| + C$
+
+134. $\int x^n \ln x \,\mathrm{d} x = \frac{1}{n+1} x^{n+1} \left( \ln x - \frac{1}{n+1} \right) + C$
+
+135. $\int (\ln x)^n \,\mathrm{d} x = x (\ln x)^n - n \int (\ln x)^{n-1} \,\mathrm{d} x$
+
+136. $\int x^m (\ln x)^n \,\mathrm{d} x = \frac{1}{m+1} x^{m+1} (\ln x)^n - \frac{n}{m+1} \int x^m (\ln x)^{n-1} \,\mathrm{d} x$
+
+## (十五) 含有双曲函数的积分
+
+137. $\int \operatorname{sh} x \,\mathrm{d} x = \operatorname{ch} x + C$
+
+138. $\int \operatorname{ch} x \,\mathrm{d} x = \operatorname{sh} x + C$
+
+139. $\int \operatorname{th} x \,\mathrm{d} x = \ln \operatorname{ch} x + C$
+
+140. $\int \operatorname{sh}^2 x \,\mathrm{d} x = -\frac{x}{2} + \frac{1}{4} \operatorname{sh} 2x + C$
+
+141. $\int \operatorname{ch}^2 x \,\mathrm{d} x = \frac{x}{2} + \frac{1}{4} \operatorname{sh} 2x + C$
+
+## (十六) 定积分
+
+142. $\int_{-\pi}^{\pi} \cos nx \,\mathrm{d} x = \int_{-\pi}^{\pi} \sin nx \,\mathrm{d} x = 0$
+
+143. $\int_{-\pi}^{\pi} \cos mx \sin nx \,\mathrm{d} x = 0$
+
+144. $\int_{-\pi}^{\pi} \cos mx \cos nx \,\mathrm{d} x = \begin{cases} 
+         0, & m \neq n, \\ 
+         \pi, & m = n. 
+     \end{cases}$
+
+145. $\int_{-\pi}^{\pi} \sin mx \sin nx \,\mathrm{d} x = \begin{cases} 
+         0, & m \neq n, \\ 
+         \pi, & m = n. 
+     \end{cases}$
+
+146. $\int_{0}^{\pi} \sin mx \sin nx \,\mathrm{d} x = \int_{0}^{\pi} \cos mx \cos nx \,\mathrm{d} x = \begin{cases} 
+         0, & m \neq n, \\ 
+         \frac{\pi}{2}, & m = n. 
+     \end{cases}$
+
+147. $I_n = \int_{0}^{\frac{\pi}{2}} \sin^n x \,dx = \int_{0}^{\frac{\pi}{2}} \cos^n x \,dx$
+
+     $I_n = \frac{n-1}{n} I_{n-2}$
+
+     $= \begin{cases} 
+         \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \, \cdots \, \cdot \frac{4}{5} \cdot \frac{2}{3} & (n \text{ 为大于 }1\text{ 的正奇数}), & I_1 = 1, \\  
+         \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \, \cdots \, \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2} & (n \text{ 为正偶数}), & I_0 = \frac{\pi}{2}.  
+     \end{cases}$