Integral calculus - Integrals of irrational functions

  1. \(\int {dx \over \sqrt {ax+b}}= {2 \over a} \sqrt {ax+b}+c\)
  2. \(\int \sqrt {ax+b}\,dx= {2 \over 3a}(ax+b)^{3/2}+c\)
  3. \(\int {adx \over \sqrt {ax+b}}= {2 (ax-2b) \over 3a^2} \sqrt{ax+b}+c\)
  4. \(\int x \sqrt {ax+b} dx= {2(3ax-2b) \over 15a^2} (ax+b)^{3/2}+c\)
  5. \(\int {dx \over (x+c) \sqrt {ax+b}}= {1 \over \sqrt {b-ac}}In \bigg | {\sqrt {ax+b}- \sqrt {b-ac} \over \sqrt {ax+b}+ \sqrt {b-ac}} \bigg |+c\),
    \(b-ac<0 .\)
  6. \(\int {dx \over (x+c) \sqrt {ax+b}}= {1 \over \sqrt {ac-b}}arctan \sqrt {ax+b \over ac-b}+c,\)
    \(b-ac<0.\)
  7. \(\int \sqrt {ax+b \over cx+d}dx={1 \over c} \sqrt {(ax+b)(cx+d)}- {ad-bc \over c \sqrt {ac} } In \bigg | \sqrt {a(cx+d)} + \sqrt {c(ax+b)} \bigg | +c \),
    \( a> 0.\)
  8. \(\int \sqrt {ax+b \over cx+d}dx={1 \over c} \sqrt {(ax+b)(cx+d)}- {ad-bc \over c \sqrt {ac} } arc tan {\sqrt {a(cx+d) \over c(ax+b)} }+c, \)
    \((a<0, c> 0).\)
  9. \(\int x^2 \sqrt {a+bx} dx= {2(8a^2-12abx+15b^2x^2) \over105 b^3} \sqrt {(a+bx)^3}+c\)
  10. \(\int {x^2 dx \over \sqrt {a+bx}}= {2(8a^2-4abx+3b^2x^2) \over 15 b^3 } \sqrt {a+bx}+c\)
  11. \(\int {dx \over x \sqrt {a+bx }} = {1 \over \sqrt a} In \bigg | {\sqrt {a+bx}- \sqrt a \over \sqrt {a+bx}+ \sqrt a}\bigg |+c,\)
    \(a>0.\)
  12. \(\int {dx \over x \sqrt {a+bx} } = {2 \over \sqrt {-a}}arctan \bigg |{a+bx \over -a } \bigg |+c, a < 0 .\)
  13. \(\int \sqrt {a-x \over b+x}dx= \sqrt {(a-x)(b+x)}+(a+b)arcsin \sqrt { x+b \over a+b}+c\)
  14. \(\int \sqrt {a+x \over b-x}dx= -\sqrt {(a+x)(b-x)}-(a+b)arcsin \sqrt { b-x \over a+b}+c\)
  15. \(\int \sqrt {1+x \over 1-x}dx= - \sqrt{1-x^2}+arcsin \, x+c\)
  16. \(\int {dx \over \sqrt {(x-a)(b-a)}}=2 arcssin \sqrt {x-a \over b-a}+c\)
  17. \(\int \sqrt {a+bx-cx^2}dx={2cx-b \over 4c} \sqrt {a+bx-cx^2}+ {b^2+4ac \over8\sqrt{c^3}}arcsin{2cx-b \over\sqrt {b^2+4ac}}+c\)
  18. \(\int {dx \over \sqrt {ax^2+bx+c}}={1 \over \sqrt a} In\bigg |2ax+b+2\sqrt {a(ax^2+bx+c)} \bigg |+c,\)
    a>0.
  19. \(\int {dx \over \sqrt {ax^2+bx+c}}=- {1 \over \sqrt a }arcsin{2ax+b \over4a} \sqrt {b^2-4ac}+c , a< 0..\)
  20. \(\int \sqrt {x^2+a^2}dx={x \over 2}\sqrt {x^2+a^2}+ {a^2 \over2}In \bigg |x+\sqrt {x^2+a^2 }\bigg |+c\)
  21. \(\int x \sqrt {x^2+a^2} dx ={1 \over 3}(x^2+a^2)^{3/2}+c\)
  22. \(\int x^2 \sqrt {x^2+a^2}dx={x \over 8}(2x^2+a^2) \sqrt {x^2+a^2}- {a^4 \over8}In \bigg |x+ \sqrt {x^2+a^2}\bigg |+c\)
  23. \(\int {\sqrt {x^2+a^2} \over x^2}dx=- {\sqrt {x^2+a^2} \over x}+In \bigg|x+ \sqrt {x^2+a^2} \bigg|+c\)
  24. \(\int {dx \over \sqrt {x^2+a^2}} = In \bigg |x+\sqrt {x^2+a^2} \bigg |+c\)
  25. \(\int \sqrt {x^2+a^2 \over x}dx=\sqrt {x^2+a^2}+a \,In \bigg | { x \over a+ \sqrt {x^2+a^2}} \bigg | +c\)
  26. \(\int {x \, dx \over \sqrt {x^2+a^2}}= In \bigg | x+ \sqrt {x^2+a^2} \bigg |+c\)
  27. \(\int {x^2dx \over \sqrt {x^2+a^2}}={x \over 2}\sqrt {x^2+a^2}- {a^2 \over2}In \bigg |{x+ \sqrt {x^2+a^2}} \bigg |+c\)
  28. \(\sqrt {dx \over x \sqrt {x^2+a^2}}={1 \over a }In \bigg| {x\over a+\sqrt {x^2+a^2}} \bigg|+c\)
  29. \(\int \sqrt {x^2-a^2}dx= {x \over 2} \sqrt {x^2-a^2}-{a^2 \over 2}In \bigg |x+ \sqrt {x^2-a^2}\bigg | +c\)
  30. \(\int \sqrt {x^2-a^2}dx = {1 \over 3}(x^2+a^2)^{3/2}+c\)
  31. \(\int {\sqrt {x^2-a^2} \over x}dx = \sqrt {x^2+a^2}+a \;arcsin {a\over x}+c\)
  32. \(\int {\sqrt {x^2-a^2} \over x^2}dx =-{ \sqrt {x^2+a^2} \over x }+ In \bigg |x+\sqrt {x^2-a^2} \bigg |+c\)
  33. \(\int {dx \over \sqrt {x^2-a^2}} = In \bigg |x+\sqrt {x^2-a^2} \bigg |+c\)
  34. \(\int {xdx \over \sqrt {x^2-a^2}} = \sqrt {x^2-a^2}+c\)
  35. \(\int {x^2dx \over \sqrt {x^2-a^2 }}= {x \over 2}\sqrt {x^2-a^2}+{a^2 \over 2 }In \bigg |x+\sqrt {x^2-a^2} \bigg|+c\)
  36. \(\int {dx \over x\sqrt {x^2-a^2}}=-{1 \over a} \, arcsin{a \over x}+c\)
  37. \(\int {dx \over (x+a)\sqrt {x^2-a^2}}={1 \over a} \sqrt {x-a \over x+a}+c\)
  38. \(\int {dx \over (x-a)\sqrt {x^2-a^2}}=-{1 \over a} \sqrt {x+a \over x-a}+c\)
  39. \(\int {dx \over x^2 \sqrt {x^2-a^2}}= {\sqrt {x^2-a^2} \over a^2x}+c\)
  40. \(\int {dx \over (x^2-a^2)^{3/2}}=- {x \over a^2\sqrt {x^2-a^2}}+c\)
  41. \(\int (x^2-a^2)^{3/2}dx=- {x \over 8} (2x^2-5a^2)\sqrt {x^2-a^2}+{3a^4 \over 8 } In \bigg |x+\sqrt {x^2-a^2} \bigg |+c\)
  42. \(\int \sqrt {a^2-x^2}dx={x \over 2} \sqrt {a^2-x^2}+{a^2 \over 2}arcsin{x \over a}+c\)
  43. \(\int x \sqrt {a^2-x^2}dx=-{1 \over 3 }(a^2-x^2)^{3/2}+c\)
  44. \(\int x^2 \sqrt {a^2-x^2}dx={x \over 8}(2x^2-a^2)\sqrt {a^2-x^2}+{a^4 \over 8 } arcsin {x \over a }+c\)
  45. \(\int {\sqrt {a^2-x^2} \over x }dx= \sqrt {a^2-x^2}+a \;In \bigg | {x \over a+\sqrt {a^2-x^2}} \bigg |+c\)
  46. \(\int {\sqrt {a^2-x^2} \over x^2}dx=-{\sqrt {a^2-x^2} \over x }-arcsin {x \over a }+c\)
  47. \(\int {dx \over \sqrt {1-x^2}}= arcssin \,x +c\)
  48. \(\int {dx \over \sqrt {a^2-x^2}}= sin \,{x \over a } +c\)
  49. \(\int {xdx \over \sqrt {a^2-x^2}}= -\sqrt {a^2-x^2} +c\)
  50. \(\int {x^2dx \over \sqrt {a^2-x^2}}= - {x \over 2} \sqrt {a^2-x^2}+{a^2 \over 2}arcsin {x \over a}+c\)
  51. \(\int {dx \over (x+a)\sqrt {a^2-x^2}}=-{1 \over 2} \sqrt {a-x \over a+x}+c\)
  52. \(\int {dx \over (x-a)\sqrt {a^2-x^2}}=-{1 \over 2} \sqrt {a+x \over a-x}+c\)
  53. \(\int {dx \over (x+b)\sqrt { a^2-x^2}}={1 \over \sqrt {b^2-a^2}}arcsin {bx+a^2 \over a(x+b)}+c, b>a .\)
  54. \(\int {dx \over (x+b)\sqrt { a^2-x^2}}={1 \over \sqrt {a^2-b^2}}In \bigg |{x+b \over \sqrt {a^2-b^2} \sqrt {a^2-x^2}+a^2+bx} \bigg | +c, b<a .\)
  55. \(\int {dx \over x^2\sqrt {a^2-x^2}}=-{\sqrt {a^2-x^2} \over a^2x}+c\)
  56. \(\int (a^2-x^2)^{3/2}dx={x \over 8}(5a^2-2x^2)\sqrt {a^2-x^2}+{3a^4 \over 8}arcsin {x \over a}+c\)
  57. \(\int {dx \over (a^2-x^2)^{3/2}}={x \over a^2 \sqrt {a^2-x^2 }}+c\)