Theorom 11.1.1 Basic Antiderivatives

\(\int k\ dx = kx + C\)

\(\int x^n\ dx = \frac{x^{n+1}}{n+1} + C, (n\ne-1)\)

\(\int e^x\ dx = e^x + C\)

\(\int a^x\ dx = \frac{a^x}{\ln(x)} + C\)

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

\(\int \cos(x)\ dx = \sin(x) + C\)

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

\(\int \tan(x)\ dx = \ln{|\cos(x)|} + C\)

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

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

\(\int \sec(x)\tan(x)\ dx = \tan(x) + C\)

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

\(\int \frac{1}{x^2+1} = \arctan(x) + C\)

\(\int \frac{1}{\sqrt{x^2+1}}\ dx = \arcsin{x} + C\)

11.1.2 Sum Rule for Antiderivatives

Given two functions \(f(x)\) and \(g(x)\) where \(k\) is a constant,

\(\int kf(x)\ dx = kF(x) + C\)

\(\int (f(x) + g(x))\ dx = F(x) + G(x)\)

11.1

  1. \(\int 5\ dx = 5x + C\)

  2. \(\int (-7x^4+8)\ dx = -\frac{7}{5}x^5 + 8x + C\)

  3. \(\int (2e^x - 4)\ dx = 2e^x - 4x + C\)

  4. \(\int (7^x - x^7)\ dx = \frac{7^x}{\ln{x}} - \frac{x^8}{8} + C\)

  5. \(\int \frac{15}{x} + x^{15}\ dx = 15\ln(x) + \frac{x^16}{16} + C\)

  6. \(\int (-3\sin(x) + tan(x))\ dx = 3\cos(x) + \sec^2(x) + C\)

  7. \(\int (\sec^2(x) - \csc^2(x)) = \tan(x) + \cot(x) + C\)

  8. \(\int (\frac{1}{x} + \frac{1}{x^2} + \frac{1}{\sqrt{x}})\ dx = \ln(x) - \frac{1}{x} + 2\sqrt{x} + C\)

  9. \(\int (\frac{17}{1+x^2} + \frac{13}{x})\ dx = 17\arctan(x) +13\ln(x) + C\)

  10. \(\int (\frac{\csc(x)\cot(x)}{4} - \frac{4}{\sqrt{1-x^2}}) = -\frac{\csc(x)}{4} - 4\arcsin(x) + C\)

  11. \(\int (2x(x^2+4)^5)\ dx = \frac{(x^2+4)^6}{6} + C\)

  12. \(\int (\frac{(\ln(x))^4}{x})\ dx = \frac{(\ln(x))^5}{5} + C\)

  13. \(\int \frac{1}{\sqrt{2x+1}}\ dx = \sqrt{2x+1} + C\)

  14. \(\int \frac{x}{\sqrt{x^2+1}}\ dx = \sqrt{x^2 + 1} + C\)

  15. \(\int x\sqrt{4-x^2}\ dx = -\frac{(4-x^2)^{3/2}}{3} + C\)

  16. \(\int \frac{\sqrt{\ln(x)}}{x}\ dx = \frac{2(\ln(x))^{3/2}}{3} + C\)

  17. \(\int (3x^2e^{x^3-1})\ dx = e^{x^3-1} + C\)

  18. \(\int (xe^{3x^2})\ dx = \frac{e^{3x^2}}{6} + C\)

  19. \(\int (2xe^{-x^2})\ dx = -e^{-x^2} + C\)

  20. \(\int (\frac{8x}{e^{x^2}})\ dx = -4e^{-x^2} + C\)

  21. \(\int (xe^{5x})\ dx = \frac{xe^{5x}}{5} - \frac{e^{5x}}{25} + C\)

  22. \(\int (xe^{-x/2})\ dx = -xe^{-x/2} + e^{-x/2} + C\)

  23. \(\int (\frac{1}{2x})\ dx = \frac{\ln(2x)}{2} + C\)

  24. \(\int (\frac{x^4}{x^5+1})\ dx = \frac{1}{5}\ln(x^5+1) + C\)

  25. \(\int (\frac{x^2}{3-x^3})\ dx = (-\frac{1}{3} \ln(3-x^3)) + C\)

  26. \(\int (\frac{1}{x\ln(x)})\ dx = \ln(x\ln(x)) - \ln(x) + C\)

  27. \(\int (\frac{e^{2x}+e^{-2x}}{e^{2x}+e^{-2x}})\ dx = \frac{1}{2}\ln(e^{2x}+e^{-2x}) + C\)

  28. \(\int (\frac{1}{x\ln(x^2)})\ dx = \frac{1}{2}\ln(x\ln(x^2)) - \frac{1}{2}\ln(x) + C\)

  29. \(\int (5x^4\sin(x^5+3))\ dx = -\cos(x^5+3) + C\)

  30. \(\int (x\cos(-2x^2)) = -\frac{1}{4} \sin(-2x^2) + C\)

  31. \(\int x\sin(5x^2)\ dx = -\frac{1}{10}\cos(5x^2) + C\)

  32. \(\int \cos(x^2)\ dx = 4\sin(x^2) + C\)

  33. \(\int 6e^{3x}\sin(e^{3x})\ dx = -6\cos(e^{3x}) + C\)

  34. \(\int (\frac{\cos(\ln(x))}{x})\ dx = \sin(\ln(x)) + C\)