Integral calculus - Reduction formulas

  1. \(\int x^ne^{mx}dx={1 \over m}x^ne^{mx}-{n \over m} \int x^{n-1}e^{mx}dx\)
  2. \(\int {e^{mx} \over x^n}dx= -{c^{mx} \over (n-1)x^{n-1}}+{m \over n-1} \int {e^{mx} \over x^{n-1}}dx, \, n \neq 1 \,.\)
  3. \(\int sinh^n \, xdx= {1 \over n}sinh^{n-1}x\, cosh \, x- { n-1 \over n} \int sin h ^{n-2}xdx\)
  4. \(\int {dx \over sinh^n \, x }= -{cosh\,x \over (n-1) sinh ^{n-1}x}- {n-2 \over n-1} \int {dx \over sinh^{n-2}x}, \, n \neq 1 \, .\)
  5. \(\int cosh ^n xdx = {1 \over n}sin h\, x cosh ^{n-1}x\, cosh \, x + {n-1 \over n} \int cosh^{n-2}xdx\)
  6. \(\int {dx \over cosh^n x }= - {sinh\,x \over (n-1)cosh^{n-1}x}+ {n-2 \over n-1} \int {dx \over cosh^{n-2}x}, n\neq 1 \,.\)
  7. \(\int sinh^n \, x \, cosh^m \, xdx= {sinh^{n+1}xcosh^{m-1}x \over n+m } + {m-1 \over n+m} \int sinh^nx \, cosh ^{m-2} \, xdx \)
  8. \(\int sinh^n \, x \, cosh^m \, xdx = {sinh ^ {n-1 } \, x \, cosh ^ {m+1}x \over n+m}- {n-1 \over n+m} \int sinh^ {n-2} x \, cosh^m\, xdx\)
  9. \(\int tanh^n , xdx = - {1 \over n-1}tanh^{n-1}x+\int tanh^{n-2}\, xdx , n \neq 1\,.\)
  10. \(\int coth^n , xdx = - {1 \over n-1}coth^{n-1}x+\int coth^{n-2}\, xdx , n \neq 1\,.\)
  11. \(\int sech^nxdx= {sech^{n-2} x\, tanh\, x \over {n-1 }}+{n-2 \over n-1}\int sech^{n-2}xdx , \, n\neq 1.\)
  12. \(\int sin^n \, xdx= -{1 \over n}sin ^{n-1}x\, cos x +{n-1 \over n } \int sin^{n-2} xdx\)
  13. \(\int {dx \over sin^n x}=-{cos x \over (n-1)sin^{n-1}x}+{n-2 \over n-1}\int {dx \over sin ^{n-2}x}\, ,n \neq 1.\)
  14. \(\int cos^n \, xdx = {1 \over n}sin \, x \, cos^{n-1} x+{n-1 \over n} \int cos ^{n-2}xdx\)
  15. \(\int {dx \over cos ^n x }= {sinx \over (n-1)cos^{n-1}x}+ {n-2 \over n-1} \int {dx \over cos^{n-2}x}\, n \neq 1.\)
  16. \(\int sin^n x \, cos^m xdx= {sin^{n+1}x\, cos ^{m-1}x \over n+m}+ {m-1 \over n+m}\ \int sin^n x \, cos ^{m-2 }xdx\)
  17. \(\int sin^n x \, cos ^m xdx = - {sin^{n-1}x \, cos ^{m+1}x \over n+m}+ {n-1 \over n+m}\int sin^{n-2}x \, cos^m xdx\)
  18. \(\int tan^n \, xdx = {1 \over n-1 }tan ^{n-1}x - \int tan^{n-2}xdx, n\neq1.\)
  19. \(\int cot^n \, xdx =- {1 \over n-1 }cot ^{n-1}x -\int cot^{n-2}xdx, n\neq1.\)
  20. \(\int sec^n xdx = {sec^{n-2} x \, tan x \over n-1}+ {n-2 \over n-1}sec^{n-2}xdx, n \neq 1.\)
  21. \(\int csc^n xdx = - {csc^{n-2} x \, cot x \over n-1}+ {n-2 \over n-1} \int csc^ {n-2}xdx, n \neq 1.\)
  22. \(\int x^n \, In^m\, xdx = {x^{n+1} In^m \; x \over n+1 } - {m \over n-1} \int x^n\, In ^{m-1} xdx\)
  23. \(\int {In^m x \over x^n}dx= - {In^m x \over (n-1)x^{n-1}}+ {m \over n-1 } \int {In^{m-1} x \over x^n }dx, n\neq 1.\)
  24. \(\int In^n \, xdx = x\, In^n \; x-n \int In ^{n-1} xdx\)
  25. \(\int x^n \, sinh\, xdx = x^n \; cosh\, x-n \int x^{n-1} cosh \, xdx \)
  26. \(\int x^n \, cosh\, xdx = x^n \; sinh\, x-n \int x^{n-1} sinh \, xdx \)
  27. \(\int x^n \, sin\, xdx = -x^n \; cos\, x+n \int x^{n-1} cos \, xdx \)
  28. \(\int x^n \, cos\, xdx = x^n \; sin\, x-n \int x^{n-1} sin \, xdx \)
  29. \(\int x^n \, sin ^{-1} \, xdx = {x^{n+1} \over n+1 }sin^{-1}x- {1 \over n+1 } \int {x^{n+1} \over \sqrt {1-x^2 }} dx\)
  30. \(\int x^n \, cos ^{-1} \, xdx = {x^{n+1} \over n+1 }cos^{-1}x+ {1 \over n+1 } \int {x^{n+1} \over \sqrt {1-x^2 }} dx\)
  31. \(\int x^n \, tan ^{-1} \, xdx = {x^{n+1} \over n+1 }tan^{-1}x -{1 \over n+1 } \int {x^{n+1} \over 1-x^2} dx\)
  32. \(\int {x^n dx \over ax^n +b } = {x \over a }- {b \over a } \int {dx \over ax^n+b }\)
  33. \(\int {dx \over (ax^2 +bx +c)^n }= {-2ax-b \over (n-1)(b^2-4ac)(ax^2+bx+c)^{n-1}}- {2(2n-3)a \over (n-1)(b^2-5ac)} \int {dx \over (ax^2+bx+c)^{n-1}}\, , n \neq1.\)
  34. \(\int {dx \over (x^2+a^2)^n}= {x \over 2(n-1)a^2 (x^2+a^2)^{n-1}}+ {2n-3 \over 2(n-1)a^2}\int {dx \over (x^2+a^2)^n} \, n \neq 1.\)
  35. \(\int {dx \over (x^2-a^2)^n}= - {x \over 2(n-1)a^2(x^2-a^2)^{n-1}}- {2n-3 \over 2(n-1)a^2}\int {dx \over (x^2-a^2)^{n-1}}, n \neq 1.\)