# Table of Integrals (see Thomas' Calculus, Fourteenth Edition.) :::{index} Integrals ::: ```{contents} :local: ``` ## 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. $\displaystyle \int x^n e^{ax} \dx = \frac{1}{a}x^n e^{ax}-\frac{n}{a} \int x^{n-1} e^{ax} \dx$