Formulas and Tables

Calculus

Integrals of Irrational Functions

Argument (independent variable): \(x\)

Real numbers: \(C\), \(a\), \(b\), \(c\)

  1. An algebraic function involving one or more radicals of polynomials is called an irrational function. Integrals of irrational functions usually contain linear, quadratic or linear fractional expressions under the radical sign.
  2. \({\large\int {\frac{{dx}}{{\sqrt {ax + b} }}}\normalsize} =\) \( {\large\frac{2}{a}\normalsize}\sqrt {ax + b} + C\)
  3. \({\large\int\normalsize} {\sqrt {ax + b}\,dx} =\) \( {\large\frac{2}{{3a}}\normalsize}{\left( {ax + b} \right)^{3/2}} + C\)
  4. \({\large\int {\frac{{xdx}}{{\sqrt {ax + b} }}}\normalsize} =\) \({\large\frac{{2\left( {ax – 2b} \right)}}{{3{a^2}}}\normalsize}\sqrt {ax + b} + C\)
  5. \({\large\int\normalsize} {x\sqrt {ax + b}\,dx} =\) \({\large\frac{{2\left( {3ax – 2b} \right)}}{{15{a^2}}}\normalsize}{\left( {ax + b} \right)^{3/2}} \) \(+\; C\)
  6. \({\large\int {\frac{{dx}}{{\left( {x + c} \right)\sqrt {ax + b} }}}\normalsize} =\) \({\large\frac{1}{{\sqrt {b – ac} }}\normalsize} \ln \left| {\large\frac{{\sqrt {ax + b} – \sqrt {b – ac} }}{{\sqrt {ax + b} + \sqrt {b – ac} }}\normalsize} \right| \) \(+\; C,\) \(b – ac \gt 0.\)
  7. \({\large\int {\frac{{dx}}{{\left( {x + c} \right)\sqrt {ax + b} }}}\normalsize} =\) \({\large\frac{1}{{\sqrt {ac – b} }}\normalsize} \arctan\sqrt {\large\frac{{ax + b}}{{ac – b}}\normalsize} \) \(+\; C,\) \(b – ac \lt 0.\)
  8. \({\large\int\normalsize} {\sqrt {{\large\frac{{ax + b}}{{cx + d}}}\normalsize} \,dx} =\) \({\large\frac{1}{c}\normalsize}\sqrt {\left( {ax + b} \right)\left( {cx + d} \right)} \) \(-\; {\large\frac{{ad – bc}}{{c\sqrt {ac} }}\normalsize}\ln \big| {\sqrt {a\left( {cx + d} \right)} }\) \( +\;{ \sqrt {c\left( {ax + b} \right)} } \big| \) \(+\; C,\) \(a \gt 0.\)
  9. \({\large\int\normalsize} {\sqrt {{\large\frac{{ax + b}}{{cx + d}}}\normalsize} \,dx} =\) \({\large\frac{1}{c}\normalsize}\sqrt {\left( {ax + b} \right)\left( {cx + d} \right)} \) \(-\; {\large\frac{{ad – bc}}{{c\sqrt {ac} }}\normalsize} \arctan\sqrt {\large\frac{{a\left( {cx + d} \right)}}{{c\left( {ax + b} \right)}}\normalsize} \) \(+\; C,\) \(a \lt 0,c \gt 0.\)
  10. \({\large\int\normalsize} {{x^2}\sqrt {ax + b}\,dx} =\) \({\large\frac{{2\left( {8{a^2} – 12abx + 15{b^2}{x^2}} \right)}}{{105{b^3}}}\normalsize}\) \(\sqrt {{{\left( {ax + b} \right)}^3}} \) \(+\;C\)
  11. \({\large\int {\frac{{{x^2}dx}}{{\sqrt {ax + b} }}}\normalsize} =\) \({\large\frac{{2\left( {8{a^2} – 4abx + 3{b^2}{x^2}} \right)}}{{15{b^3}}}\normalsize}\) \(\sqrt {ax + b} \) \(+\; C\)
  12. \({\large\int {\frac{{dx}}{{x\sqrt {a + bx} }}}\normalsize} =\) \({\large\frac{1}{{\sqrt a }}\normalsize}\ln \left| {\large\frac{{\sqrt {a + bx} – \sqrt a }}{{\sqrt {a + bx} + \sqrt a }}\normalsize} \right| \) \(+\; C,\) \(a \gt 0.\)
  13. \({\large\int {\frac{{dx}}{{x\sqrt {a + bx} }}}\normalsize} =\) \( {\large\frac{2}{{\sqrt { – a} }}\normalsize} \arctan\left| {\large\frac{{a + bx}}{{ – a}}\normalsize} \right| + C,\) \(a \lt 0.\)
  14. \({\large\int\normalsize} {\sqrt {{\large\frac{{a – x}}{{b + x}}}\,dx}\normalsize} =\) \(\sqrt {\left( {a – x} \right)\left( {b + x} \right)} \) \(+\; \left( {a + b} \right)\arcsin \sqrt {\large\frac{{x + b}}{{a + b}}\normalsize} \) \(+\; C\)
  15. \({\large\int\normalsize} {\sqrt {{\large\frac{{a + x}}{{b – x}}}\,dx}\normalsize} =\) \(-\sqrt {\left( {a + x} \right)\left( {b – x} \right)} \) \(-\; \left( {a + b} \right)\arcsin \sqrt {\large\frac{{b – x}}{{a + b}}\normalsize} \) \(+\; C\)
  16. \({\large\int\normalsize} {\sqrt {{\large\frac{{1 + x}}{{1 – x}}}\normalsize}\,dx} =\) \( – \sqrt {1 – {x^2}} \) \(+\; \arcsin x + C\)
  17. \({\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {\left( {x – a} \right)\left( {b – a} \right)} }}\normalsize}} =\) \({ 2\arcsin \sqrt {\large\frac{{x – a}}{{b – a}}} }\normalsize + C\)
  18. \({\large\int\normalsize} {\sqrt {a + bx – c{x^2}}\,dx} =\) \({\large\frac{{2cx – b}}{{4c}}\normalsize}\sqrt {a + bx – c{x^2}} \) \(+\;{\large\frac{{{b^2} – 4ac}}{{8\sqrt {{c^3}} }}\normalsize}\arcsin {\large\frac{{2cx – b}}{{\sqrt {{b^2} + 4ac} }}\normalsize}\) \(+\; C\)
  19. \({\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {a{x^2} + bx + c} }}}\normalsize} =\) \( {\large\frac{1}{{\sqrt a }}\normalsize}\ln \big| {2ax + b }\) \(+{\; 2\sqrt {a\left( {a{x^2} + bx + c} \right)} } \big|\) \(+\; C,\) \(a \gt 0.\)
  20. \({\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {a{x^2} + bx + c} }}}\normalsize} =\) \( -{\large\frac{1}{{\sqrt { – a} }}\normalsize} \arcsin{\large\frac{{2ax + b}}{{4a}}\normalsize}\) \( \sqrt {{b^2} – 4ac} \) \(+\; C,\) \(a \lt 0.\)
  21. \({\large\int\normalsize} {\sqrt {{x^2} + {a^2}} dx} =\) \({\large\frac{x}{2}\normalsize}\sqrt {{x^2} + {a^2}} \) \(+\;{\large\frac{{{a^2}}}{2}\normalsize}\ln\left| {x + \sqrt {{x^2} + {a^2}} } \right| \) \(+\; C\)
  22. \({\large\int\normalsize} {x\sqrt {{x^2} + {a^2}} dx} =\) \({\large\frac{1}{3}\normalsize}{\left( {{x^2} + {a^2}} \right)^{3/2}} + C\)
  23. \({\large\int\normalsize} {{x^2}\sqrt {{x^2} + {a^2}} dx} =\) \({\large\frac{x}{8}\normalsize}\left( {2{x^2} + {a^2}} \right)\sqrt {{x^2} + {a^2}} \) \(-\;{\large\frac{{{a^4}}}{8}\normalsize}\ln \left| {x + \sqrt {{x^2} + {a^2}} } \right|\) \(+\; C\)
  24. \({\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} + {a^2}} }}{{{x^2}}}\normalsize} dx} =\) \( – {\large\frac{{\sqrt {{x^2} + {a^2}} }}{x}\normalsize} \) \(+\; \ln \left| {x + \sqrt {{x^2} + {a^2}} } \right| + C\)
  25. \({\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {{x^2} + {a^2}} }}}\normalsize} =\) \(\ln \left| {x + \sqrt {{x^2} + {a^2}} } \right| + C\)
  26. \({\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} + {a^2}} }}{x}\normalsize}\,dx} =\) \(\sqrt {{x^2} + {a^2}} \) \(+\; a\ln \left| {\large\frac{x}{{a + \sqrt {{x^2} + {a^2}} }}\normalsize} \right| + C\)
  27. \({\large\int\normalsize} {\large{\frac{{xdx}}{{\sqrt {{x^2} + {a^2}} }}}\normalsize} =\) \( \sqrt {{x^2} + {a^2}} + C\)
  28. \({\large\int\normalsize} {\large{\frac{{{x^2}dx}}{{\sqrt {{x^2} + {a^2}} }}}\normalsize} =\) \({\large\frac{x}{2}\normalsize}\sqrt {{x^2} + {a^2}} \) \(-\; {\large\frac{{{a^2}}}{2}\normalsize}\ln \left| {x + \sqrt {{x^2} + {a^2}} } \right|\) \(+\; C\)
  29. \({\large\int\normalsize} {\large{\frac{{dx}}{{x\sqrt {{x^2} + {a^2}} }}}\normalsize} =\) \({\large\frac{1}{a}\normalsize}\ln \left| {\large\frac{x}{{a + \sqrt {{x^2} + {a^2}} }}\normalsize} \right| + C\)
  30. \({\large\int\normalsize} {\sqrt {{x^2} – {a^2}} dx} =\) \({\large\frac{x}{2}\normalsize}\sqrt {{x^2} – {a^2}} \) \(-\;{\large\frac{{{a^2}}}{2}\normalsize}\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|\) \(+\; C\)
  31. \({\large\int\normalsize} {x\sqrt {{x^2} – {a^2}} dx} =\) \({\large\frac{1}{3}\normalsize}{\left( {{x^2} – {a^2}} \right)^{3/2}} + C\)
  32. \({\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} – {a^2}} }}{x}\normalsize} dx} =\) \(\sqrt {{x^2} – {a^2}} \) \(+\; a\arcsin {\large\frac{a}{x}\normalsize} + C\)
  33. \({\large\int\normalsize} {{\large\frac{{\sqrt {{x^2} – {a^2}} }}{{{x^2}}}\normalsize} dx} =\) \( -{\large\frac{{\sqrt {{x^2} – {a^2}} }}{x}\normalsize} \) \(+\; \ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|\) \(+\; C\)
  34. \({\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {{x^2} – {a^2}} }}}\normalsize} =\) \(\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right| + C\)
  35. \({\large\int\normalsize} {\large{\frac{{xdx}}{{\sqrt {{x^2} – {a^2}} }}}\normalsize} =\) \(\sqrt {{x^2} – {a^2}} + C\)
  36. \({\large\int\normalsize} {\large{\frac{{{x^2}dx}}{{\sqrt {{x^2} – {a^2}} }}}\normalsize} =\) \({\large\frac{x}{2}\normalsize}\sqrt {{x^2} – {a^2}} \) \(+\;{\large\frac{{{a^2}}}{2}\normalsize}\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|\) \(+\; C\)
  37. \({\large\int\normalsize} {\large{\frac{{dx}}{{x\sqrt {{x^2} – {a^2}} }}}\normalsize} =\) \( – {\large\frac{1}{a}\normalsize}\arcsin {\large\frac{a}{x}\normalsize} + C\)
  38. \({\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x + a} \right)\sqrt {{x^2} – {a^2}} }}}\normalsize} =\) \({\large\frac{1}{a}\normalsize}\sqrt {\large\frac{{x – a}}{{x + a}}\normalsize} + C\)
  39. \({\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x – a} \right)\sqrt {{x^2} – {a^2}} }}}\normalsize} =\) \(-{\large\frac{1}{a}\normalsize}\sqrt {\large\frac{{x + a}}{{x – a}}\normalsize} + C\)
  40. \({\large\int\normalsize} {\large{\frac{{dx}}{{{x^2}\sqrt {{x^2} – {a^2}} }}}\normalsize} =\) \({\large\frac{{\sqrt {{x^2} – {a^2}} }}{{{a^2}x}}\normalsize} + C\)
  41. \({\large\int\normalsize} {\large{\frac{{dx}}{{{{\left( {{x^2} – {a^2}} \right)}^{3/2}}}}}\normalsize} =\) \( – {\large\frac{x}{{{a^2}\sqrt {{x^2} – {a^2}} }}\normalsize} + C\)
  42. \({\large\int\normalsize} {{{\left( {{x^2} – {a^2}} \right)}^{3/2}}dx} =\) \(-\;{\large\frac{x}{8}\normalsize}\left( {2{x^2} – 5{a^2}} \right)\sqrt {{x^2} – {a^2}} \) \(+\;{\large\frac{{3{a^4}}}{8}\normalsize}\ln \left| {x + \sqrt {{x^2} – {a^2}} } \right|\) \(+\; C\)
  43. \({\large\int\normalsize} {\sqrt {{a^2} – {x^2}} dx} =\) \( {\large\frac{x}{2}\normalsize}\sqrt {{a^2} – {x^2}} \) \(+\;{\large\frac{{{a^2}}}{2}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C\)
  44. \({\large\int\normalsize} {x\sqrt {{a^2} – {x^2}} dx} =\) \( – {\large\frac{1}{3}\normalsize}{\left( {{a^2} – {x^2}} \right)^{3/2}} + C\)
  45. \({\large\int\normalsize} {{x^2}\sqrt {{a^2} – {x^2}} dx} =\) \({\large\frac{x}{8}\normalsize}\left( {2{x^2} – {a^2}} \right)\sqrt {{a^2} – {x^2}} \) \(+\;{\large\frac{{{a^4}}}{8}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C\)
  46. \({\large\int\normalsize} {{\large\frac{{\sqrt {{a^2} – {x^2}} }}{x}\normalsize} dx} =\) \(\sqrt {{a^2} – {x^2}} \) \(+\; a\ln \left| {\large\frac{x}{{a + \sqrt {{a^2} – {x^2}} }}\normalsize} \right| + C\)
  47. \({\large\int\normalsize} {{\large\frac{{\sqrt {{a^2} – {x^2}} }}{{{x^2}}}\normalsize} dx} =\) \( – {\large\frac{{\sqrt {{a^2} – {x^2}} }}{x}\normalsize} \) \(-\; \arcsin {\large\frac{x}{a}\normalsize} + C\)
  48. \({\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {1 – {x^2}} }}}\normalsize} =\) \( \arcsin x + C\)
  49. \({\large\int\normalsize} {\large{\frac{{dx}}{{\sqrt {{a^2} – {x^2}} }}}\normalsize} =\) \( \arcsin {\large\frac{x}{a}\normalsize} + C\)
  50. \({\large\int\normalsize} {\large{\frac{{xdx}}{{\sqrt {{a^2} – {x^2}} }}}\normalsize} =\) \( – \sqrt {{a^2} – {x^2}} + C\)
  51. \({\large\int\normalsize} {\large{\frac{{{x^2}dx}}{{\sqrt {{a^2} – {x^2}} }}}\normalsize} =\) \( – {\large\frac{x}{2}\normalsize}\sqrt {{a^2} – {x^2}} \) \(+\;{\large\frac{{{a^2}}}{2}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C\)
  52. \({\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x + a} \right)\sqrt {{a^2} – {x^2}} }}}\normalsize} =\) \( – {\large\frac{1}{2}\normalsize}\sqrt {\large\frac{{a – x}}{{a + x}}\normalsize} + C\)
  53. \({\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x – a} \right)\sqrt {{a^2} – {x^2}} }}}\normalsize} =\) \( – {\large\frac{1}{2}\normalsize}\sqrt {\large\frac{{a + x}}{{a – x}}\normalsize} + C\)
  54. \({\large\int\normalsize} {\large\frac{{dx}}{{\left( {x + b} \right)\sqrt {{a^2} – {x^2}} }}\normalsize} =\) \({\large\frac{1}{{\sqrt {{b^2} – {a^2}} }}\normalsize}\arcsin {\large\frac{{bx + {a^2}}}{{a\left( {x + b} \right)}}\normalsize}\) \(+\; C,\) \(b \gt a.\)
  55. \({\large\int\normalsize} {\large{\frac{{dx}}{{\left( {x + b} \right)\sqrt {{a^2} – {x^2}} }}}\normalsize} =\) \({\large\frac{1}{{\sqrt {{a^2} – {b^2}} }}\normalsize}\) \(\ln\left| {\large\frac{{x + b}}{{\sqrt {{a^2} – {b^2}} \sqrt {{a^2} – {x^2}} + {a^2} + bx}}\normalsize} \right|\) \(+\;C,\) \(b \gt a.\)
  56. \({\large\int\normalsize} {\large{\frac{{dx}}{{{x^2}\sqrt {{a^2} – {x^2}} }}}\normalsize} =\) \( – {\large\frac{{\sqrt {{a^2} – {x^2}} }}{{{a^2}x}}\normalsize} + C\)
  57. \({\large\int\normalsize} {{{\left( {{a^2} – {x^2}} \right)}^{3/2}}dx} =\) \({\large\frac{x}{8}\normalsize}\left( {5{a^2} – 2{x^2}} \right)\sqrt {{a^2} – {x^2}} \) \(+\;{\large\frac{{3{a^4}}}{8}\normalsize}\arcsin {\large\frac{x}{a}\normalsize} + C\)
  58. \({\large\int\normalsize} {\large{\frac{{dx}}{{{{\left( {{a^2} – {x^2}} \right)}^{3/2}}}}}\normalsize} =\) \({\large\frac{x}{{{a^2}\sqrt {{a^2} – {x^2}} }}\normalsize} + C\)