1. $$\int \frac{1+x^{2}}{1+x^{4}} \mathrm{~d} x$$
  2. $$\int \frac{\mathrm{d} x}{(1+x)\left(1+x^{2}\right)}$$
  3. $$\int \frac{\mathrm{d} x}{\left(1+x^{3}\right)\left(1+x^{2}\right)}$$
  4. $$\int \frac{\mathrm{d} x}{\sqrt[3]{1-x^{3}}}$$
  5. $$\int \frac{\mathrm{d} x}{\lambda+\sqrt{1-x^{2}}}$$
  6. $$\int \frac{\mathrm{d} x}{a \sin x+b \cos x}$$
  7. $$\int \ln (\sqrt{1+x}+\sqrt{1-x}) \mathrm{d} \mathrm{x}$$
  8. $$\int \ln (\sqrt{1-x}-\sqrt{x}) \mathrm{d} x$$
  9. $$\int \frac{\sin x+\cos x}{1-\sin x \cos x} \mathrm{~d} x$$
  10. $$\int \frac{1}{\sqrt{1+e^{x}}+\sqrt{1-e^{x}}} \mathrm{~d} x$$
  11. $$\int \frac{x+\sqrt{1-x^{2}}}{1-x \sqrt{1-x^{2}}} \mathrm{~d} x$$
  12. $$\int \frac{\mathrm{d} x}{\sec x+\csc x+\cot x+\tan x}$$
  13. $$\int x^{4} \sqrt{\frac{1-x}{1+x}} \mathrm{~d} x$$
  14. $$\int x \sqrt[4]{\frac{1-x}{1+x}} d x$$
  15. $$\int \sqrt{x+\sqrt{x}} \mathrm{~d} x$$
  16. $$\int \frac{1-\ln x}{(x-\ln x)^{2}} \mathrm{~d} x$$
  17. $$\int \tan x \tan (a+x) \mathrm{d} x$$
  18. $$\int e^{\frac{x}{2}} \frac{\cos x}{\sqrt{\cos x+\sin x}} \mathrm{~d} x$$
  19. $$\int \frac{\mathrm{dx}}{\sqrt{\tan ^{2} x+2}}$$
  20. $$\int \sqrt{\tan ^{2} x+2} \mathrm{dx}$$
  21. $$\int \arcsin \sqrt{\frac{a-x}{a+x}} \mathrm{dx}$$
  22. $$\int \frac{1}{x} \sqrt{\frac{1+x}{1-x}} \mathrm{~d} x$$
  23. $$\int \frac{x+\sin x}{1+\cos x} \mathrm{dx}$$
  24. $$\int \frac{x+\cos x}{1+\sin x} \mathrm{~d} x$$
  25. $$\int \ln x^{2} \arcsin x \mathrm{~d} x$$
  26. $$\int \frac{\sin (\ln x)}{x^{2}} \mathrm{~d} x$$
  27. $$\int \frac{x^{2} \ln x}{\left(x^{2}+1\right)^{\frac{3}{2}}} \mathrm{~d} x$$
  28. $$\int \frac{\sin x \cos x}{\sin ^{4} x+\cos ^{4} x} \mathrm{dx}$$
  29. $$\int \frac{x^{2}}{(x \sin x+\cos x)^{2}} \mathrm{~d} x$$
  30. $$\int \frac{2 x+\sin 2 x}{(\cos x-x \sin x)^{2}} \mathrm{~d} x$$
  31. $$\int \frac{x+1}{x\left(1+x e^{x}\right)} \mathrm{d} x$$
  32. $$\int \frac{1}{\sqrt{\operatorname{th} x+1}} \mathrm{~d} x$$
  33. $$\int \sqrt{\frac{x^{2}}{2}-\frac{2}{x^{2}}} \mathrm{~d} x$$
  34. $$\int \arctan \left(1-\frac{1}{x}\right) \mathrm{d} x$$
  35. $$\int \frac{\mathrm{d} x}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$$
  36. $$\int \sqrt{x \sqrt[3]{x \sqrt[4]{x \sqrt[5]{x \ldots}}}} d x$$
  37. $$\frac{\sum_{k=0}^{n}(k+1) x^{k}}{\sum_{k=0}^{n+1} x^{k}} \mathrm{~d} x$$
  38. $$\int \frac{x^{n}}{\sum_{k=0}^{n} \frac{x^{k}}{k !}} \mathrm{d} x$$
  1. $$\int \frac{\cos (\sin t)+\cos ^{2} t}{1+\sin t \sin (\sin t)} \mathrm{d} x$$
  2. $$\int \frac{x^{2}-2 \sinh ^{2} x}{\left(x^{2}-1\right) \sinh (2 x)+2 x} \mathrm{~d} x$$
  3. $$\int\left[\frac{x \sin (\ln (x+1))}{(1+\sqrt{1+x})}\right]^{2} \mathrm{~d} x$$
  4. $$\int \frac{x+1}{\exp \{2 / x\}} \exp \left\{x e^{-\frac{1}{x}}\right\} \mathrm{d} x$$
  5. $$\int \frac{\ln \left(\sqrt{1+x^{2}}-x\right) \mathrm{d} x}{\sqrt{1+x^{2}} \sqrt{x+\sqrt{1+x^{2}}}}$$
  6. $$\int \frac{x^{2} \mathrm{~d} x}{\left(x^{2}-4\right) \sin x+4 x \cos x}$$
  7. $$\int \frac{x^{3}-2}{\left(x^{3}+1\right)^{2}} \sqrt{x^{3}-x^{2}+1} \mathrm{~d} x$$
  8. $$\int \frac{\sin ^{3} x}{\sin ^{3} x+\cos ^{3} x} \mathrm{~d} x$$
  9. $$\int \frac{\mathrm{d} x}{(1+x)\left(1+x^{2}\right)\left(1+x^{3}\right)}$$
  10. $$\int \frac{\cos 2 x-\tan x \cdot \cot (\tan x)}{\sin 2 x-\tan (\tan x) \ln \left(\cos ^{2} x\right)} \mathrm{d} x$$
  11. $$\int \frac{d x}{\sin ^{6} x+\cos ^{6} x}$$
  12. $$\int \frac{\sin 4 x}{\sin ^{8} x+\cos ^{8} x} \mathrm{~d} x$$
  13. $$\int \frac{\cos x \mathrm{~d} x}{\sqrt{2+\sin 2 x}}$$
  14. $$\int \frac{\mathrm{d} x}{(x-a)^{4}+(x-b)^{4}}$$
  15. $$\int \sqrt{x+a^{2} \sqrt{x-a}} \mathrm{~d} x$$
  16. $$\int \frac{x^{2} \mathrm{~d} x}{a^{2}+x^{2}+\sqrt{a^{2}+x^{2}}}$$
  17. $$\int \frac{x+x^{3}+x^{5}+2 x^{7}}{\exp \left\{x^{2}-x^{4}\right\}} \mathrm{d} x$$
  18. $$\int \frac{x \cos x \mathrm{~d} x}{3+4 \sin x-\cos 2 x}$$
  19. $$\int \frac{x \sin x+\cos x}{(x+\cos x)^{2}} \mathrm{~d} x$$
  20. $$\int \frac{(\sin x+1)^{2}+x \cos x}{(x \sin x-\cos x)^{2}} \mathrm{~d} x$$
  21. $$\int e^{x} \frac{1+\sin x}{1+\cos x} \mathrm{~d} x$$
  22. $$\int \frac{x(1+x) \mathrm{d} x}{\left(e^{x}+x+1\right)^{2}}$$
  23. $$\int \frac{\tan x-\cos x}{\left(2+e^{\sin x} \cos x\right)^{2}} \mathrm{~d} x$$
  24. $$\int \frac{(2 \cosh x-1) \mathrm{d} x}{\left(e^{x} \sin x+\cos x\right)\left(\cos x-e^{-x} \sin x\right)}$$
  25. $$\int e^{-x^{2}}\left(\left(x^{2}+2^{-1}\right)^{-2}-2\right) \mathrm{d} x$$
  26. $$\int \frac{e^{x}\left((x-1)^{2}-2\right)}{\left(x^{2}-1\right)^{2}} \mathrm{~d} x$$
  27. $$\int \frac{x-2}{\sqrt{e^{x}-x^{2}}} \mathrm{~d} x$$
  28. $$\int \frac{\mathrm{d} \mathrm{x}}{(1+x \tan x)^{2}}$$
  29. $$\int \frac{\mathrm{d} x}{(\sin x+a \sec x)^{2}}$$
  30. $$\int \frac{\cos x(2020 x+2019 \sin x \cos x)}{(x \sin x+\cos x)^{3}} \mathrm{~d} x$$
  31. $$\int \frac{(x+1) e^{x}}{\sqrt{a^{2}-e^{x}}} \mathrm{~d} x$$
  32. $$\int \sqrt{\tan x} \mathrm{dx}$$
  33. $$\int \sqrt{\tan x+1} \mathrm{dx}$$
  34. $$\int \sqrt{\frac{1+\sqrt{2} \sin x}{1+\sqrt{2} \cos x}} \mathrm{~d} x$$
  35. $$\int \sqrt[3]{\frac{1+\sin x}{1-\sin x}} \mathrm{~d} x$$
  36. $$\int \frac{1}{\sqrt{\tan x}} \mathrm{~d} x$$
  37. $$\int \frac{1}{\sqrt{\tan x+1}} \mathrm{~d} x$$
  38. $$\int \frac{\mathrm{d} x}{(1+\sin x)^{n}}$$
  39. $$\int\left(\frac{\arctan x}{\arctan x-x}\right)^{2} \mathrm{dx}$$
  40. $$\int \frac{2 x \arctan x-1}{\arctan ^{2} x} \mathrm{~d} x$$
  41. $$\int \arctan \left(\frac{x^{2}-x-1}{1-2 x}\right) \mathrm{d} x \quad( 暴力不能解决一切 \sim)$$
  42. $$\int \arctan \frac{x^{3}-x^{2}-4 x-1}{x^{3}+4 x^{2}+x-1} \mathrm{~d} x$$
  43. $$\int \arccos \left(7 x^{2}-\sqrt{49 x^{4}+1-50 x^{2}}\right) \ln x \mathrm{~d} x$$
  44. $$\int \frac{x^{4}-1}{x^{8}+1} \sqrt{1+x^{4}} \mathrm{~d} x$$
  45. $$\int \frac{\arcsin \sqrt{x} \arccos \sqrt{x}}{\sqrt{1-x}} \mathrm{~d} x$$
  46. $$\int \frac{\arctan x}{x^{2}+x^{-2}+2} \mathrm{~d} x$$
  47. $$\int \frac{2 x \arctan x(\arctan x+x)}{\left(1+x^{2}\right)^{2}} \mathrm{~d} x$$
  48. $$\int \frac{x}{\sqrt{1-x^{2}}} \ln \frac{x}{\sqrt{1-x^{2}}} \mathrm{~d} x$$
  49. $$\int \frac{\ln (x+m)-\ln (x+n)}{(x+m)^{2}(x+n)^{2}} \mathrm{~d} x$$
  50. $$\int \frac{(x+m) \ln (x+m)+(x+n) \ln (x+n)}{(x+m)^{2}(x+n)^{2}} \mathrm{~d} x$$
  51. $$\int \frac{1}{\ln x}\left(\frac{1}{\ln ^{2} x}-\frac{1}{2}\right) d x$$
  52. $$\int \frac{\frac{\ln x}{x}+\ln ^{2} x}{e^{-2 x}+\ln ^{2} x} \mathrm{~d} x$$
  53. $$\int \frac{\ln ^{2} x+1-\left(\ln ^{2} x+1\right)^{-\frac{1}{2}}}{2 x \ln x} \mathrm{dx}$$
  54. $$\int \frac{\ln \left(x^{2}\right)+1-\left(\ln \left(x^{2}\right)+1\right)^{-\frac{1}{2}}}{2 x \ln x} \mathrm{dx}$$
  55. $$\int \csc x \ln \left(\tan \frac{x}{2}\right) \mathrm{d} x$$
  56. $$\int \csc ^{2} x \ln (\cos x+\sqrt{\cos 2 x}) \mathrm{d} x$$
  57. $$\int \sqrt{\frac{x+\sqrt{x}}{x-\sqrt{x}}} \mathrm{~d} x$$
  58. $$\int \frac{x+1+\ln x}{(x+1)^{2}+(x \ln x)^{2}} \mathrm{~d} x$$
  59. $$\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} \mathrm{~d} x$$
  60. $$\int \frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}} \mathrm{dx}$$
  61. $$\int \sqrt[3]{x(1-x)(1+x)} \mathrm{d} x$$
  62. $$\int\left(\frac{1}{x}-\sqrt{x}\right)^{-\frac{1}{2}} \mathrm{~d} x$$
  63. $$\int \frac{d x}{\sin 4 x+4 \sin x}$$
  64. $$\int \frac{\sin x \cos x}{\sqrt{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}} \mathrm{~d} x$$
  65. $$\int \frac{\mathrm{d} x}{x^{n}\left(1+x^{2}\right)}$$
  66. $$\int \frac{\sin x \mathrm{~d} x}{\sqrt{1+\sin 2 x}}$$
  67. $$\int \sqrt{\frac{\sin (x+\xi)}{\sin (x-\xi)}} \mathrm{d} x$$
  68. $$\int \sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}} \mathrm{d} x$$
  69. $$\int \frac{1}{\sqrt{2}+\sqrt{1-x}+\sqrt{1+x}} \mathrm{~d} x$$
  70. $$\int\left(x^{x^{2}+2}+1\right)\left(\ln x^{2}+1\right) \frac{\mathrm{d} x}{x}$$
  71. $$\int \frac{\mathrm{d} x}{\sqrt{\left(x+x^{-1}\right)^{2}-12}}$$
  72. $$\int \frac{x(2-x) e^{x} \cos 2 x+e^{2 x}-x^{4}}{\left(e^{x} \cos x+x^{2} \sin x\right) \sqrt{x^{4}-e^{2 x}}} \frac{\mathrm{d} x}{\sqrt{\cos 2 x}}$$
  73. $$\int \exp \{\sec x\} \frac{(1+\sin x)(1+\tan x)}{\cos ^{2} x} \mathrm{~d} x$$
  74. $$\int \frac{x}{\sqrt{e^{-x}-e^{-2 x}}} \mathrm{~d} x$$
  75. $$\int \frac{\mathrm{d} x}{\sqrt[3]{e^{-2 x}+e^{-3 x}}-\sqrt{e^{-x}+e^{-2 x}}}$$
  76. $$\int x\left[\frac{1}{x}\right] \mathrm{d} x$$
  77. $$\int[x]|\sin \pi x| \mathrm{d} x$$
  78. $$\int \frac{x\left(x^{2}+x \tan x+1\right)}{(x \tan x-1)^{2}} \mathrm{~d} x$$
  79. $$\int \frac{x\left(x^{2}+x \tan x+1\right)}{(x \tan x+1)^{2}} \mathrm{~d} x$$
  80. $$\int \frac{\sec ^{2} x}{(\sec x+\tan x)^{n}} \mathrm{~d} x$$
  81. $$\int \sqrt{\csc ^{2} x+\cot ^{2} x} \mathrm{~d} x$$
  82. $$\int \frac{x \mathrm{~d} x}{2(\sec x+\tan x)-\cos x}$$
  83. $$\int \frac{\mathrm{d} x}{\sin x+\cos x+\sin x \cos x}$$
  84. $$\int \frac{\mathrm{d} x}{\cos (x-1) \cos (x-2) \cos (x-3)}$$
  85. $$\int \frac{(x \cos x-\sin x) \mathrm{d} x}{(x+a \sin x)(x+b \sin x)}$$
  86. $$\int \frac{\sqrt{\cos 2 x}}{\sin x} \mathrm{~d} x$$
  87. $$\int \frac{1}{\cos x \sqrt{\cos 2 x}} \mathrm{~d} x$$
  88. $$\int \frac{\mathrm{d} x}{\sqrt{\sin 2 x \cos ^{2} x}}$$
  89. $$\int \frac{\sin ^{2} \frac{x}{2} \tan \frac{x}{2}}{\sqrt{\cos x+\cos ^{2} x+\cos ^{3} x}} \mathrm{~d} x$$
  90. $$\int \frac{\sec x}{\sqrt{\sin (2 x+\xi)+\sin \xi}} \mathrm{d} x$$
  91. $$\int \frac{\sin (\operatorname{arccot} x)}{\cot (\arcsin x)} \mathrm{d} x$$
  92. $$\int \sqrt{\frac{\csc x-\cot x}{\csc x+\cot x}} \frac{\sec x}{\sqrt{1+2 \sec x}} \mathrm{~d} x$$
  93. $$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} \mathrm{~d} x$$
  94. $$\int e^{x} \frac{2 x+(x+1) \sin 2 x}{1+\cos 2 x} \mathrm{~d} x$$
  95. $$\int \frac{\exp \{1 / \ln x\}}{x \ln ^{3} x} \mathrm{~d} x$$
  96. $$\int \frac{x+x^{-1}+1}{\sqrt{\left(x^{2}+x-1 / 3\right)^{2}+4 x / 3}} \mathrm{~d} x$$
  97. $$\int \frac{x^{n} \mathrm{~d} x}{\sqrt{\sum_{k=0}^{n}(k+1) x^{k}}}$$
    $$\int \frac{2 n ! \sin x+x^{n}}{e^{x}+\sin x+\cos x+\sum_{k=0}^{n} \frac{x^{k}}{k !}} \mathrm{d} x$$
  98. 设
    $$J_{n}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (n t-x \sin t) \mathrm{d} t,$$
    (结果用$J_n(x)$表示)，求
    $$\int \sin (x) J_{0}(x) \mathrm{d} x$$

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