Table of Integrals

(see Thomas’ Calculus, Fourteenth Edition.)

Basic Forms

1. \(\displaystyle \int k \dx = kx+C, \quad k \in \R \)

2. \(\displaystyle \int x^n \dx = \frac{x^{n+1}}{n+1}+C, \quad n \neq -1\)

3. \(\displaystyle \int \frac{1}{x} \dx = \ln |x|+C\)

4. \(\displaystyle \int e^x \dx = e^x + C\)

5. \(\displaystyle \int a^x \dx = \frac{a^x}{\ln a} + C, \quad (a>0, a\neq -1) \)

6. \(\displaystyle \int \sin x \dx = -\cos x + C \)

7. \(\displaystyle \int \cos x \dx = \sin x + C \)

8. \(\displaystyle \int \sec^2 x \dx = \tan x + C \)

9. \(\displaystyle \int \csc^2 x \dx = -\cot x + C\)

10. \(\displaystyle \int \sec x \tan x \dx = \sec x + C\)

11. \(\displaystyle \int \csc x \cot x \dx = - \csc x + C\)

12. \(\displaystyle \int \tan x \dx = \ln |\sec x| +C \)

13. \(\displaystyle \int \cot x \dx = \ln |\sin x| + C\)

14. \(\displaystyle \int \sinh x \dx = \cosh x + C\)

15. \(\displaystyle \int \cosh x \dx = \sinh x + C\)

16. \(\displaystyle \int \frac{1}{\sqrt{a^2-x^2}} \dx = \sin^{-1}\biggl(\frac{x}{a}\biggl) + C\)

17. \(\displaystyle \int \frac{1}{\sqrt{a^2+x^2}} \dx = \frac{1}{a}\tan^{-1}\biggl(\frac{x}{a}\biggl)+C\)

18. \(\displaystyle \int \frac{1}{x\sqrt{x^2-a^2}} \dx = \frac{1}{a}\sec^{-1}\biggl| \frac{x}{a} \biggr| + C\)

19. \(\displaystyle \int \frac{1}{\sqrt{a^2+x^2}} \dx = \sinh^{-1}\biggl(\frac{x}{a}\biggl) + C, \quad (a>0)\)

20. \(\displaystyle \int \frac{1}{\sqrt{x^2-a^2}} \dx = \cosh^{-1}\biggl(\frac{x}{a}\biggl) + C, \quad (x>a>0)\)

Forms involving \(ax+b\)

21. \(\displaystyle \int (ax+b)^n \dx = \frac{(ax+b)^{n+1}}{a(n+1)}+C, \quad n \neq -1\)

22. \(\displaystyle \int x(ax+b)^n \dx = \frac{(ax+b)^n}{a^2} \left[\frac{ax+b}{n+2}-\frac{b}{n+1}\right]+C, \quad n\neq -1, -2\)

23. \(\displaystyle \int (ax+b)^{-1} \dx = \frac{1}{a} \ln \left|ax+b\right| + C\)

24. \(\displaystyle \int x(ax+b)^{-1} \dx = \frac{x}{a} - \frac{b}{a^2} \ln |ax+b| + C\)

25. \(\displaystyle \int x(ax+b)^{-2} \dx = \frac{1}{a^2} \left[\ln |ax+b| + \frac{b}{ax+b}\right] + C\)

26. \(\displaystyle \int \frac{1}{x(ax+b)} \dx = \frac{1}{b} \ln \left| \frac{x}{ax+b} \right| +C\)

27. \(\displaystyle \int (\sqrt{ax+b})^n \dx = \frac{2}{a} \frac{(\sqrt{ax+b})^{n+2}}{n+2} + C, \quad n\neq -2\)

28. \(\displaystyle \int \frac{\sqrt{ax+b}}{x} \dx = 2 \sqrt{ax+b} + b \int \frac{1}{x\sqrt{ax+b}} \dx\)

29. • \(\displaystyle \int \frac{1}{x\sqrt{ax+b}} \dx = \frac{1}{\sqrt{b}} \ln \left| \frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \right| + C\)

    • \(\displaystyle \int \frac{1}{x\sqrt{ax-b}} \dx = \frac{2}{\sqrt{b}} \tan^{-1} \sqrt{\frac{ax-b}{b}}+ C\)

30. \(\displaystyle \int \frac{\sqrt{ax+b}}{x} \dx = - \frac{\sqrt{ax+b}}{x} + \frac{a}{2} \int \frac{1}{x\sqrt{ax+b}} \dx + C\)

31. \(\displaystyle \int \frac{1}{x^2 \sqrt{ax+b}} \dx = - \frac{\sqrt{ax+b}}{bx} - \frac{a}{2b} \int \frac{1}{x\sqrt{ax+b}} \dx + C\)

Forms involving \(a^2+x^2\)

32. \(\displaystyle \int \frac{1}{a^2+x^2} \dx = \frac{1}{a} \tan^{-1}\frac{x}{a} +C \)

33. \(\displaystyle \int \frac{1}{(a^2+x^2)^2} \dx= \frac{x}{2a^2(a^2+x^2)} + \frac{1}{2a^3} \tan^{-1} \frac{x}{a} + C\)

34. \(\displaystyle \int \frac{1}{\sqrt{a^2+x^2}} \dx = \sinh^{-1} \frac{x}{a} + C = \ln (x+\sqrt{a^2+x^2} + C)\)

35. \(\displaystyle \int \sqrt{a^2+x^2} \dx = \frac{x}{2}\sqrt{a^2+x^2} + \frac{a^2}{2} \ln (x+\sqrt{a^2+x^2}) + C\)

36. \(\displaystyle \int x^2 \sqrt{a^2+x^2} \dx = \frac{x}{8}(a^2+2x^2)\sqrt{a^2+x^2} - \frac{a^4}{8} \ln (x+\sqrt{a^2+x^2}) + C\)

37. \(\displaystyle \int \frac{\sqrt{a^2+x^2}}{x} \dx = \sqrt{a^2+x^2} - a \ln \left| \frac{a+\sqrt{a^2+x^2}}{x} \right| + C\)

38. \(\displaystyle \int \frac{\sqrt{a^2+x^2}}{x^2} \dx = \ln (x+\sqrt{a^2+x^2}) - \frac{\sqrt{a^2+x^2}}{x} + C\)

39. \(\displaystyle \int \frac{x^2}{\sqrt{a^2+x^2}} \dx = -\frac{a^2}{2}\ln (x+\sqrt{a^2+x^2}) + \frac{x\sqrt{a^2+x^2}}{2}+C\)

40. \(\displaystyle \int \frac{1}{x\sqrt{a^2+x^2}} \dx = -\frac{1}{a} \ln \left| \frac{a+\sqrt{a^2+x^2}}{x} \right| + C\)

41. \(\displaystyle \int \frac{1}{x^2 \sqrt{a^2+x^2}} \dx = -\frac{\sqrt{a^2+x^2}}{a^2x} + C\)

Forms involving \(a^2-x^2\)

42. \(\displaystyle \int \frac{1}{a^2-x^2} \dx = \frac{1}{2a} \ln \left|\frac{x+a}{x-a}\right|+C\)

43. \(\displaystyle \int \frac{1}{(a^2-x^2)^2} = \frac{x}{2a^2(a^2-x^2)} + \frac{1}{4a^3} \ln \left| \frac{x+a}{x-a} \right| + C\)

44. \(\displaystyle \int \frac{1}{\sqrt{a^2-x^2}} \dx= \sin^{-1} \frac{x}{a}+C\)

45. \(\displaystyle \int \sqrt{a^2-x^2} \dx = \frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin^{-1} \frac{x}{a}+C\)

46. \(\displaystyle \int x^2 \sqrt{a^2-x^2} \dx = \frac{a^4}{8} \sin^{-1} \frac{x}{a} - \frac{1}{8} x \sqrt{a^2-x^2} (a^2-2x^2) +C\)

47. \(\displaystyle \int \frac{\sqrt{a^2-x^2}}{x} \dx= \sqrt{a^2-x^2} - a \ln \left| \frac{a+\sqrt{a^2-x^2}}{x} \right| + C\)

48. \(\displaystyle \int \frac{\sqrt{a^2-x^2}}{x^2} \dx = -\sin^{-1}\frac{x}{a} - \frac{\sqrt{a^2-x^2}}{x} + C\)

49. \(\displaystyle \int \frac{1}{\sqrt{a^2-x^2}}\dx = \frac{a^2}{2} \sin^{-1} \frac{x}{a} - \frac{1}{2}x \sqrt{a^2-x^2} + C\)

50. \(\displaystyle \int \frac{1}{x\sqrt{a^2-x^2}} \dx = - \frac{1}{a} \ln \left| \frac{a+\sqrt{a^2-x^2}}{x} \right|+C\)

51. \(\displaystyle \int \frac{1}{x^2 \sqrt{a^2-x^2}} \dx = - \frac{\sqrt{a^2-x^2}}{a^2x} +C\)

Forms involving \(x^2-a^2\)

52. \(\displaystyle \int \frac{1}{\sqrt{x^2-a^2}} \dx = \ln \left| x+\sqrt{x^2-a^2}\right|+C\)

53. \(\displaystyle \int \sqrt{x^2-a^2} \dx = \frac{x}{2} \sqrt{x^2-a^2} - \frac{a^2}{2} \ln \left| x+\sqrt{x^2-a^2}\right| + C\)

54. \(\displaystyle \int (\sqrt{x^2-a^2})^n \dx = \frac{x(\sqrt{x^2-a^2})^n}{n+1} - \frac{na^2}{n+1} \int (\sqrt{x^2-a^2})^{n-2} \dx, \quad n\neq -1 \)

55. \(\displaystyle \int \frac{1}{(\sqrt{x^2-a^2})^n} \dx= \frac{x(\sqrt{x^2-a^2})^{2-n}}{(2-n)a^2} - \frac{n-3}{(n-2)a^2} \int \frac{1}{(\sqrt{x^2-a^2})^{n-2}}, \quad n\neq 2\)

Exponential and Logarithmic Forms

109. \(\displaystyle \int e^{ax} \dx = \frac{1}{a}e^{ax} + C\)

110. \(\displaystyle \int b^{ax} \dx = \frac{1}{a} \frac{b^{ax}}{\ln b} + C, \quad b>0, ~ b\neq 1\)

111. \(\displaystyle \int x e^{ax} \dx = \frac{e^{ax}}{a^2}(ax-1) + C\)

112. \(\displaystyle \int x^n e^{ax} \dx = \frac{1}{a}x^n e^{ax}-\frac{n}{a} \int x^{n-1} e^{ax} \dx\)