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   Интегралы от иррациональных функций
Аргумент (независимая переменная): \(x\)
Действительные числа: \(C\), \(a\), \(b\), \(c\)
  1. Алгебраическая функция называется иррациональной, если переменная находится под знаком корня. Интегралы от иррациональных функций, как правило, содержат линейные, квадратичные или дробно-линейные выражения под знаком корня.

  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. \(\large\int {\frac{{dx}}{{\left( {x + c} \right)\sqrt {ax + b} }}}\normalsize = \large\frac{1}{{\sqrt {b - ac} }}\normalsize \ln \left| {\large\frac{{\sqrt {ax + b} - \sqrt {b - ac} }}{{\sqrt {ax + b} + \sqrt {b - ac} }}\normalsize} \right| + C,\;\;b - ac > 0.\) 

  7. \(\large\int {\frac{{dx}}{{\left( {x + c} \right)\sqrt {ax + b} }}}\normalsize = \large\frac{1}{{\sqrt {ac - b} }}\normalsize \arctan\sqrt {\large\frac{{ax + b}}{{ac - b}}\normalsize} + C,\;\;b - ac < 0.\) 

  8. \(\large\int {\sqrt {\frac{{ax + b}}{{cx + d}}}\normalsize dx} = \large\frac{1}{c}\normalsize\sqrt {\left( {ax + b} \right)\left( {cx + d} \right)} - \large\frac{{ad - bc}}{{c\sqrt {ac} }}\normalsize\ln \left| {\sqrt {a\left( {cx + d} \right)} + \sqrt {c\left( {ax + b} \right)} } \right| + C,\;\;a > 0.\) 

  9. \(\large\int {\sqrt {\frac{{ax + b}}{{cx + d}}}\normalsize dx} = \large\frac{1}{c}\normalsize\sqrt {\left( {ax + b} \right)\left( {cx + d} \right)} - \large\frac{{ad - bc}}{{c\sqrt {ac} }}\normalsize \arctan\sqrt {\large\frac{{a\left( {cx + d} \right)}}{{c\left( {ax + b} \right)}}\normalsize} + C,\;\;a < 0,c > 0.\) 

  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. \(\large\int {\frac{{dx}}{{x\sqrt {a + bx} }}}\normalsize = \large\frac{1}{{\sqrt a }}\normalsize\ln \left| {\large\frac{{\sqrt {a + bx} - \sqrt a }}{{\sqrt {a + bx} + \sqrt a }}\normalsize} \right| + C,\;\;a > 0.\) 

  13. \(\large\int {\frac{{dx}}{{x\sqrt {a + bx} }}}\normalsize = \large\frac{2}{{\sqrt { - a} }}\normalsize \arctan\left| {\large\frac{{a + bx}}{{ - a}}\normalsize} \right| + C,\;\;a < 0.\) 

  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. \(\large\int {\frac{{dx}}{{\sqrt {\left( {x - a} \right)\left( {b - a} \right)} }}\normalsize = 2\arcsin \sqrt {\large\frac{{x - a}}{{b - a}}} }\normalsize + C\) 

  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. \(\large\int {\frac{{dx}}{{\sqrt {a{x^2} + bx + c} }}}\normalsize = \large\frac{1}{{\sqrt a }}\normalsize\ln \left| {2ax + b + 2\sqrt {a\left( {a{x^2} + bx + c} \right)} } \right| + C,\;\;a > 0.\) 

  20. \(\large\int {\frac{{dx}}{{\sqrt {a{x^2} + bx + c} }}}\normalsize = - \large\frac{1}{{\sqrt { - a} }}\normalsize \arcsin\large\frac{{2ax + b}}{{4a}}\normalsize\sqrt {{b^2} - 4ac} + C,\;\;a < 0.\) 

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

  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. \(\large\int {\frac{{xdx}}{{\sqrt {{x^2} + {a^2}} }}}\normalsize = \sqrt {{x^2} + {a^2}} + C\) 

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

  29. \(\large\int {\frac{{dx}}{{x\sqrt {{x^2} + {a^2}} }}}\normalsize = \large\frac{1}{a}\normalsize\ln \left| {\large\frac{x}{{a + \sqrt {{x^2} + {a^2}} }}\normalsize} \right| + C\) 

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

  35. \(\large\int {\frac{{xdx}}{{\sqrt {{x^2} - {a^2}} }}}\normalsize = \sqrt {{x^2} - {a^2}} + C\) 

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

  37. \(\large\int {\frac{{dx}}{{x\sqrt {{x^2} - {a^2}} }}}\normalsize = - \large\frac{1}{a}\normalsize\arcsin \large\frac{a}{x}\normalsize + C\) 

  38. \(\large\int {\frac{{dx}}{{\left( {x + a} \right)\sqrt {{x^2} - {a^2}} }}}\normalsize = \large\frac{1}{a}\normalsize\sqrt {\large\frac{{x - a}}{{x + a}}\normalsize} + C\) 

  39. \(\large\int {\frac{{dx}}{{\left( {x - a} \right)\sqrt {{x^2} - {a^2}} }}}\normalsize = -\large\frac{1}{a}\normalsize\sqrt {\large\frac{{x + a}}{{x - a}}\normalsize} + C\) 

  40. \(\large\int {\frac{{dx}}{{{x^2}\sqrt {{x^2} - {a^2}} }}}\normalsize = \large\frac{{\sqrt {{x^2} - {a^2}} }}{{{a^2}x}}\normalsize + C\) 

  41. \(\large\int {\frac{{dx}}{{{{\left( {{x^2} - {a^2}} \right)}^{3/2}}}}}\normalsize = - \large\frac{x}{{{a^2}\sqrt {{x^2} - {a^2}} }}\normalsize + C\) 

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

  49. \(\large\int {\frac{{dx}}{{\sqrt {{a^2} - {x^2}} }}}\normalsize = \arcsin \large\frac{x}{a}\normalsize + C\) 

  50. \(\large\int {\frac{{xdx}}{{\sqrt {{a^2} - {x^2}} }}}\normalsize = - \sqrt {{a^2} - {x^2}} + C\) 

  51. \(\large\int {\frac{{{x^2}dx}}{{\sqrt {{a^2} - {x^2}} }}}\normalsize = - \large\frac{x}{2}\normalsize\sqrt {{a^2} - {x^2}} + \large\frac{{{a^2}}}{2}\normalsize\arcsin \large\frac{x}{a}\normalsize + C\) 

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

  53. \(\large\int {\frac{{dx}}{{\left( {x - a} \right)\sqrt {{a^2} - {x^2}} }}}\normalsize = - \large\frac{1}{2}\normalsize\sqrt {\large\frac{{a + x}}{{a - x}}\normalsize} + C\) 

  54. \(\large\int\normalsize {\large\frac{{dx}}{{\left( {x + b} \right)\sqrt {{a^2} - {x^2}} }}\normalsize} = \large\frac{1}{{\sqrt {{b^2} - {a^2}} }}\normalsize\arcsin \large\frac{{bx + {a^2}}}{{a\left( {x + b} \right)}}\normalsize + C,\;\;b > a.\) 

  55. \(\large\int {\frac{{dx}}{{\left( {x + b} \right)\sqrt {{a^2} - {x^2}} }}}\normalsize = \large\frac{1}{{\sqrt {{a^2} - {b^2}} }}\normalsize \ln\left| {\large\frac{{x + b}}{{\sqrt {{a^2} - {b^2}} \sqrt {{a^2} - {x^2}} + {a^2} + bx}}\normalsize} \right| + C,\;\;b < a.\) 

  56. \(\large\int {\frac{{dx}}{{{x^2}\sqrt {{a^2} - {x^2}} }}}\normalsize = - \large\frac{{\sqrt {{a^2} - {x^2}} }}{{{a^2}x}}\normalsize + C\) 

  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. \(\large\int {\frac{{dx}}{{{{\left( {{a^2} - {x^2}} \right)}^{3/2}}}}}\normalsize = \large\frac{x}{{{a^2}\sqrt {{a^2} - {x^2}} }}\normalsize + C\) 



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