Integrals of Irrational Functions

Posted In: Mathematics

An irrational function is a function whose analytic expression has the independent variable $x$ under the root symbol.

  1. $∫{dx}/√{ax+b}=2/a√{ax+b}+c$
  2. $∫√{ax+b}\ dx=2/{3a}(ax+b)^{3/2}+c$
  3. $∫{xdx}/√{ax+b}={2(ax-2b)}/{3a^2}√{ax+b}+c$
  4. $∫x√{ax+b}\ dx={2(3ax-2b)}/{15a^2}(ax+b)^{3/2}+c$
  5. $∫{dx}/{(x+c)√{ax+b}}=1/√{b-ac}\ ln|{√{ax+b}-√{b-ac}}/{√{ax+b}+√{b-ac}}|+c_1$,
    $b-ac>0$.
  6. $∫{dx}/{(x+c)√{ax+b}}=1/√{ac-b}\ tan^{-1}{√{ax+b}/√{ac-b}}+c_1$,
    $b-ac<0$.
  7. $∫√{{ax+b}/{cx+d}}\ dx$
    $=1/c√{(ax+c)(cx+d)}-{ad-bc}/{c√{ac}}\ ln|√{a(cx+d)}+√{c(ax+b)}|+c_1,a>0$
  8. $∫√{{ax+b}/{cx+d}}\ dx$
    $=1/c√{(ax+c)(cx+d)}-{ad-bc}/{c√{ac}}\ tan^{-1}{√{{a(cx+d)}/{c(ax+b)}}+c_1,(a<0,c>0)$
  9. $∫x^2√{a+bx}\ dx={2(8a^2-12abx+15b^2x^2)}/{105b^3}√(a+bx)^3+c$
  10. $∫x^2/√{a+bx}\ dx={2(8a^2-4abx+3b^2x^2)}/{15b^3}√(a+bx)+c$
  11. $∫{dx}/{x√{a+bx}}=1/√a\ ln|{√{a+bx}-√a}/{√{a+bx}+√a}|+c, a>0$
  12. $∫{dx}/{x√{a+bx}}=2/√{-a}\ tan^{-1}({a+bx}/{-a})+c, a<0$
  13. $∫√{{a-x}/{b+x}}\ dx=√{(a-x)(b+x)}+(a+b)sin^{-1}√{{x+b}/{a+b}}+c$
  14. $∫√{{a+x}/{b-x}}\ dx=-√{(a+x)(b-x)}-(a+b)sin^{-1}√{{b-x}/{a+b}}+c$
  15. $∫√{{1+x}/{1-x}}\ dx=-√{1-x^2}+sin^{-1}x+c$
  16. $∫{dx}/√{(x-a)(b-a)}=2sin^{-1}√{(x-a)/(b-a)}+c$
  17. $∫√{a+bx-cx^2}\ dx={2cx-b}/{4c}√{a+bx-cx^2}+$
    $+{b^2-4ac}/{8√c^3}\ sin^{-1}{{2cx-b}/√{b^2+4ac}}+c_1$
  18. $∫{dx}/√{ax^2+bx+c}=1/√a\ ln|2ax+b+2√{a(ax^2+bx+c)}|+c, a>0$
  19. $∫{dx}/√{ax^2+bx+c}=-1/√a\ sin^{-1}({2ax+b}/{4a}√{b^2-4ac})+c, a<0$
  20. $∫√{x^2+a^2}\ dx=x/2√{x^2+a^2}+a^2/2\ ln|x+√{x^2+a^2}|+c$
  21. $∫x√{x^2+a^2}\ dx=1/3(x^2+a^2)^{3/2}+c$
  22. $∫x^2√{x^2+a^2}\ dx=x/8(2x^2+a^2)√{x^2+a^2}-a^4/8\ ln|x+√{x^2+a^2}|+c$
  23. $∫{dx}/√{x^2+a^2}=ln|x+√{x^2+a^2}|+c$
  24. $∫√{x^2+a^2}/x\ dx=√{x^2+a^2}+a\ ln|x/{a+√{x^2+a^2}}|+c$
  25. $∫{x\ dx}/√{x^2+a^2}=√{x^2+a^2}+c$
  26. $∫{x^2\ dx}/√{x^2+a^2}=x/2√{x^2+a^2}-a^2/2\ ln|x+√{x^2+a^2}|+c$
  27. $∫{dx}/{x√{x^2+a^2}}=1/a\ ln|x/{a+√{x^2+a^2}}|+c$
  28. $∫√{x^2-a^2}\ dx=x/2√{x^2-a^2}-a^2/2\ ln|x+√{x^2-a^2}|+c$
  29. $∫x√{x^2-a^2}\ dx=1/3(x^2-a^2)^{3/2}+c$
  30. $∫√{x^2-a^2}/x\ dx=√{x^2-a^2}+a\ sin^{-1}a/x+c$
  31. $∫√{x^2-a^2}/x^2\ dx=-√{x^2-a^2}/x+ln\|x+√{x^2-a^2}|+c$
  32. $∫{dx}/√{x^2-a^2}=ln|x+√{x^2-a^2}|+c$
  33. $∫{x\ dx}/√{x^2-a^2}=√{x^2-a^2}+c$
  34. $∫{x^2\ dx}/√{x^2-a^2}=x/2√{x^2-a^2}+a^2/2\ ln|x+√{x^2-a^2}|+c$
  35. $∫{dx}/{x√{x^2-a^2}}=-1/a\ sin^{-1}a/x+c$
  36. $∫{dx}/{(x+a)√{x^2-a^2}}=1/a√{{x-a}/{x+a}}+c$
  37. $∫{dx}/{(x-a)√{x^2-a^2}}=-1/a√{{x+a}/{x-a}}+c$
  38. $∫{dx}/{x^2√{x^2-a^2}}=√{x^2-a^2}/{a^2x}+c$
  39. $∫(x^2-a^2)^{-3/2}\ dx=∫{dx}/(x^2-a^2)^{3/2}=-x/{a^2√{x^2-a^2}}+c$
  40. $∫(x^2-a^2)^{3/2}\ dx=-x/8(2x^2-5a^2)√{x^2-a^2}+{3a^4}/8\ ln|x+√{x^2-a^2}|+c$
  41. $∫√{a^2-x^2}\ dx=x/2√{a^2-x^2}+a^2/2\ sin^{-1}x/a+c$
  42. $∫x√{a^2-x^2}\ dx=-1/3(a^2-x^2)^{3/2}+c$
  43. $∫x^2√{a^2-x^2}\ dx=x/8(2x^2-a^2)√{a^2-x^2}+a^4/8\ sin^{-1}x/a+c$
  44. $∫√{a^2-x^2}/x\ dx=√{a^2-x^2}+a\ ln|x/{a+√{a^2-x^2}}|+c$
  45. $∫√{a^2-x^2}/x^2\ dx=-√{a^2-x^2}/x-sin^{-1}x/a+c$
  46. $∫{dx}/√{a^2-x^2}=sin^{-1}x/a+c$
  47. $∫{dx}/√{1^2-x^2}=sin^{-1}x+c$
  48. $∫{xdx}/√{a^2-x^2}=-√{a^2-x^2}+c$
  49. $∫{x^2dx}/√{a^2-x^2}=-x/2√{a^2-x^2}+a^2/2sin^{-1}x/a+c$
  50. $∫{dx}/{(x+a)√{a^2-x^2}}=-1/2√{{a-x}/{a+x}}+c$
  51. $∫{dx}/{(x-a)√{a^2-x^2}}=-1/2√{{a+x}/{a-x}}+c$
  52. $∫{dx}/{(x+b)√{a^2-x^2}}=1/√{b^2-a^2}\ sin^{-1}{{bx+a^2}/{a(x+b)}}|+c, b>a$
  53. $∫{dx}/{(x+b)√{a^2-x^2}}=1/√{a^2-b^2}\ ln|{x+b}/{√{a^2-b^2}√{a^2-x^2}+a^2}|+c, b<a$
  54. $∫{dx}/{x^2√{a^2-x^2}}=-√{a^2-x^2}/{a^2x}+c$
  55. $∫(a^2-x^2)^{3/2}\ dx=x/8(5a^2-2x^2)√{a^2-x^2}+{3a^4}/8\ sin^{-1}x/a+c$
  56. $∫(a^2-x^2)^{-3/2}\ dx=x/{a^2√{a^2-x^2}}+c$

Post Tags: Integration


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