Reduction Formulas

Posted In: Mathematics
  1. $∫x^ne^{mx}\ dx=1/mx^ne^{mx}-n/m∫x^{n-1}e^{mx}\ dx$
  2. $∫{e^{mx}}/{x^n}\ dx=-e^{mx}/{(n-1)x^{n-1}}+m/{n-1}∫e^{mx}/{x^{n-1}}\ dx, n≠1$
  3. $∫sin\ h^nx\ dx=1/nsin\ h^{n-1}x\ cos\ hx-{n-1}/n∫sin\ h^{n-2}x\ dx$
  4. $∫{dx}/{sin\ h^nx}=-{cos\ hx}/{(n-1)sin\ h^{n-1}x}-{n-2}/{n-1}∫{dx}/{sin\ h^{n-2}x},n≠1$
  5. $∫cos\ h^nx\ dx=1/nsin\ hx\ cos\ h^{n-1}x\ cos\ hx+{n-1}/n∫cos\ h^{n-2}x\ dx$
  6. $∫{dx}/{cos\ h^nx}=-{sin\ hx}/{(n-1)cos\ h^{n-1}x}+{n-2}/{n-1}∫{dx}/{cos\ h^{n-2}x},n≠1$
  7. $∫sin\ h^nx\ cos\ h^mx\ dx={sin\ h^{n+1}x\ cos\ h^{m-1}x}/{n+m}$
    $+{m-1}/{n+m}∫sin\ h^n\ cos\ h^{m-2}x\ dx$
  8. $∫sin\ h^nx\ cos\ h^mx\ dx={sin\ h^{n-1}x\ cos\ h^{m+1}x}/{n+m}$
    $-{n-1}/{n+m}∫sin\ h^{n-2}\ cos\ h^mx\ dx$
  9. $∫tan\ h^nx\ dx=-1/(n-1)tan\ h^{n-1}x+∫tan\ h^{n-2}x\ dx,n≠1$
  10. $∫cot\ h^nx\ dx=-1/(n-1)cot\ h^{n-1}x+∫cot\ h^{n-2}x\ dx,n≠1$
  11. $∫sec\ h^nx\ dx={sec\ h^{n-2}x\ tan\ hx}/(n-1)+{n-2}/{n-1}∫sec\ h^{n-2}x\ dx,n≠1$
  12. $∫sin^nx\ dx=-1/nsin^{n-1}x\ cos\ x+{n-1}/n∫sin^{n-2}x\ dx$
  13. $∫{dx}/{sin^nx}=-{cos\ x}/{(n-1)sin^{n-1}x}+{n-2}/{n-1}∫{dx}/{sin^{n-2}x},n≠1$
  14. $∫cos^nx\ dx=1/nsin\ x\ cos^{n-1}x+{n-1}/n∫cos^{n-2}x\ dx$
  15. $∫{dx}/{cos^nx}={sin\ x}/{(n-1)cos^{n-1}x}+{n-2}/{n-1}∫{dx}/{cos^{n-2}x},n≠1$
  16. $∫sin^nx\ cos^mx\ dx={sin^{n+1}x\ cos^{m-1}x}/{n+m}+{m-1}/{n+m}∫sin^nx\ cos^{m-2}x\ dx$
  17. $∫sin^nx\ cos^mx\ dx=-{sin^{n-1}x\ cos^{m+1}x}/{n+m}+{n-1}/{n+m}∫sin^{n-2}x\ cos^mx\ dx$
  18. $∫tan^nx\ dx=1/{n-1}tan^{n-1}x-∫tan^{n-2}x\ dx,n≠1$
  19. $∫cot^nx\ dx=-1/{n-1}cot^{n-1}x-∫cot^{n-2}x\ dx,n≠1$
  20. $∫sec^nx\ dx={sec^{n-2}x\ tan\ x}/{n-1}+{n-2}/{n-1}∫sec^{n-2}x\ dx,n≠1$
  21. $∫cosec^nx\ dx=-{sec^{n-2}x\ cot\ x}/{n-1}+{n-2}/{n-1}∫cosec^{n-2}x\ dx,n≠1$
  22. $∫x^n\ ln^mx\ dx={x^{n+1}\ ln^mx}/{n+1}-m/{n+1}∫x^{n-1}\ ln^mx\ dx$
  23. $∫{ln^mx\ dx}/{x^n}=-{ln^mx}/{(n-1)x^{n-1}}+m/{n-1}∫{ln^{m-1}x\ dx}/{ln^nx},n≠1$
  24. $∫ln^nx\ dx=x\ ln^n\ x-n∫ln^{n-1}x\ dx$
  25. $∫x^n\ sin\ hx\ dx=x^n\ cos\ hx-n∫x^{n-1}\ cos\ hx\ dx$
  26. $∫x^n\ cos\ hx\ dx=x^n\ sin\ hx-n∫x^{n-1}\ sin\ hx\ dx$
  27. $∫x^n\ sin\ x\ dx=-x^n\ cos\ x+n∫x^{n-1}\ cos\ x\ dx$
  28. $∫x^n\ cos\ x\ dx=x^n\ sin\ x-n∫x^{n-1}\ sin\ x\ dx$
  29. $∫x^n\ sin^{-1} x\ dx={x^{n+1}}/{n+1}\ sin^{-1}\ x-1/{n+1}∫{x^{n+1}}/√{1-x^2}\ dx$
  30. $∫x^n\ cos^{-1} x\ dx={x^{n+1}}/{n+1}\ cos^{-1}\ x+1/{n+1}∫{x^{n+1}}/√{1-x^2}\ dx$
  31. $∫x^n\ tan^{-1} x\ dx={x^{n+1}}/{n+1}\ tan^{-1}\ x-1/{n+1}∫{x^{n+1}}/{1+x^2}\ dx$
  32. $∫{x^n\ dx}/{ax^n+b}=x/a-b/a∫{dx}/{ax^n+b}$
  33. $∫{dx}/(ax^2+bx+c)^n$
    $={-2ax-b}/{(n-1)(b^2-4ac)(ax^2+bx+c)^{n-1}}-{2(2n-3)a}/{(n-1)(b^2-4ac)}∫{dx}/(ax^2+bx+c)^{n-1},n≠1$
  34. $∫{dx}/(x^2+a^2)^n$
    $=x/{2(n-1)a^2(x^2+a^2)^{n-1}}+{2n-3}/{2(n-1)a^2}∫{dx}/(x^2+a^2)^{n-1},n≠1$
  35. $∫{dx}/(x^2-a^2)^n$
    $=-x/{2(n-1)a^2(x^2-a^2)^{n-1}}-{2n-3}/{2(n-1)a^2}∫{dx}/(x^2-a^2)^{n-1},n≠1$

Post Tags: Integration


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