Integral calculus - Integrals of trigonometric functions

  1. \(\int sin \, xdx= - cos \,x +c\)
  2. \(\int cos \, xdx= - sin \,x +c\)
  3. \(\int sin^2x \,dx= {x \over2}-{1 \over 4}sin \,2x+c\)
  4. \(\int cos^2x \,dx= {x \over2}+{1 \over 4}sin \,2x+c\)
  5. \(\int sin^3 \, x \,dx ={1 \over 3}cos^3 \, x- cos \, x+c={1 \over12}cos3x-{3 \over4}cos x+c\)
  6. \(\int cos^3 \, x \,dx =sin \, x- {1 \over 3}sin^3 \, x+c={1 \over12}sin3x+{3 \over4}sin x+c\)
  7. \(\int {dx \over sin \, x}= \int csc \, x \, dx = In \bigg |tan {x \over2} \bigg| +c\)
  8. \(\int {dx \over cos \, x}= \int sec \, x \, dx = In \bigg |tan \bigg ({x \over2 }+ { \pi \over 4 }\bigg) \bigg| +c\)
  9. \(\int { dx \over sin^2 x } = \int csc^2 x \, dx=-cot \,x+c\)
  10. \(\int { dx \over cos^2 x } = \int sec^2 x \, dx=tan \,x+c\)
  11. \(\int { dx \over sin ^3 \, x } = \int csc^3 \, x \,dx =-{cos \, x \over 2 sin ^2 \,x }+ { 1 \over 2 }In \bigg |tan {x \over 2} \bigg |+c\)
  12. \(\int { dx \over cos ^3 \, x } = \int sec^3 \, x \,dx =-{sin \, x \over 2 cos ^2 \,x }+ { 1 \over 2 }In \bigg |tan \bigg ( {x \over 2}+ {\pi \over 4}\bigg )\bigg|+c\)
  13. \(\int sin^2 \, x. cos \, x \, dx = {1 \over 3} sin^3x +c\)
  14. \(\int sin \, x. cos^2 \, x \, dx =- {1 \over 3}cos^3x +c\)
  15. \(\int sin ^2x. cos^2 x \, dx ={x \over 8}-{1 \over 32}sin4x+c\)
  16. \(\int tan \,xdx = -In |cos x| +c\)
  17. \(\int {sin \,x \over cos^2 x}dx = {1 \over cos \, x} +c =sce \, x +c\)
  18. \(\int { sin^2 \,x \over cos \, x }dx = In\bigg |tan\bigg({x \over 2}+{\pi \over 4} \bigg ) \bigg |-sin \,x+c\)
  19. \(\int tan ^2xdx=tan \, x-x+c\)
  20. \(\int cot \, x dx=In \bigg |sin \, x\bigg |+c\)
  21. \(\int {cos \,x \over sin^2 x }dx= -{1 \over sin\, x}+c= -csc\, x+c\)
  22. \(\int {cos^2 \,x \over sin x }dx= In \bigg |tan {x \over 2 }\bigg |+cos \, x +c\)
  23. \(\int cot ^2 \, x dx = - cot \, x -x + c\)
  24. \(\int {dx \over cos \,x. sin \,x }= In \bigg |tan \, x \bigg |+c\)
  25. \(\int {dx \over sin^2 \,x. cos \, x}= -{1 \over sin \, x}+In \bigg |tan \bigg ({x \over 2} + {\pi \over 2} \bigg ) \bigg |+c\)
  26. \(\int {dx \over sin \,x. cos^2 \, x}= {1 \over cos\, x}+In \bigg |tan {x \over 2} \bigg |+c\)
  27. \(\int {dx \over sin ^2 x. cos^2 x}= tan x-cotx+c\)
  28. \(\int sin \, mx . sin \, nx \, dx = -{sin (m+n)x \over 2(m+n) }+{sin (m-n)x \over 2(m-n) }+c , m^2 \neq n^2.\)
  29. \(\int sin \, mx . cos \, nx \, dx = -{cos (m+n)x \over 2(m+n) }-{cos (m-n)x \over 2(m-n) }+c , m^2 \neq n^2.\)
  30. \(\int cos \, mx . cos \, nx \, dx = {sin (m+n)x \over 2(m+n) }+{sin (m-n)x \over 2(m-n) }+c , m^2 \neq n^2.\)
  31. \(\int sec \, x. tan \, x dx= sec \, x +c\)
  32. \(\int csc \, x. cot \, x dx=-csc \, x +c\)
  33. \(\int sin \, x. cos^n \, x dx =- { cos^{n+1} x\over n+1 }+c\)
  34. \(\int sin^n \, x. cos \, x dx = {sin^{n+1} x\over n+1 }+c\)
  35. \(\int arcsin \, x \, dx = x \, arc sin \, x+ \sqrt {1- x^2}+c\)
  36. \(\int arccos \, x \, dx = x \, arc cos \, x- \sqrt {1- x^2}+c\)
  37. \(\int arctan \, x \, dx = x \, arc tan \, x- {1 \over 2}In({x^2+ 1})+c\)
  38. \(\int arccot \, x \, dx = x \, arc cot \, x+ {1 \over 2}In({x^2+ 1})+c\)