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. \(\int arccot \, x \, dx = x \, arc cot \, x+ {1 \over 2}In({x^2+ 1})+c\)

Indefinite Integral

Integrals of Rational Functions

Integrals of Irrational Functions

Integrals of Trigonometric Functions

Integrals of Hyperbolic Functions

Integrals of Exponential and Logarithmic Functions

Reduction Formulas

Definite Integral

Improper Integral

Double Integral