Integrals of Trigonometric Functions

Posted In: Mathematics
  1. $∫sin\ x\ dx=-cos\ x+c$
  2. $∫cos\ x\ dx=sin\ x+c$
  3. $∫sin^2\ x \ dx=x/2-1/4sin\ 2x+c$
  4. $∫cos^2\ x \ dx=x/2+1/4sin\ 2x+c$
  5. $∫sin^3\ x \ dx=1/3cos^3\ x-cos\ x+c=1/12cos\ 3x-3/4cos\ x+c$
  6. $∫cos^3\ x \ dx=sin\ x-1/3sin^3\ x+c=1/12sin\ 3x+3/4sin\ x+c$
  7. $∫{dx}/{sin\ x}=∫cosec\ x\ dx=ln|tan{x/2}|+c$
  8. $∫{dx}/{cos\ x}=∫sec\ x\ dx=ln|tan(π/4+x/2)|+c$
  9. $∫{dx}/{sin^2\ x}=∫cosec^2\ x\ dx=-cot\ x+c$
  10. $∫{dx}/{cos^2\ x}=∫sec^2\ x\ dx=tan\ x+c$
  11. $∫{dx}/{sin^3\ x}=∫cosec^3\ x\ dx=-{cos\ x}/{2sin^2\ x}+1/2\ ln|tan\ x/2|+c$
  12. $∫{dx}/{cos^3\ x}=∫sec^3\ x\ dx={sin\ x}/{2cos^2\ x}+1/2\ ln|tan(π/4+x/2)|+c$
  13. $∫sin\ x\ cos\ x\ dx=-1/4cos\ 2x+c$
  14. $∫sin^2\ x\ cos\ x\ dx=1/3sin^3\ x+c$
  15. $∫sin\ x\ cos^2\ x\ dx=-1/3cos^3\ x+c$
  16. $∫sin^2\ x\ cos^2\ x\ dx=x/8-1/32sin4\ x+c$
  17. $∫tan\ x\ dx=-ln\ cos\ x+c$
  18. $∫{sin\ x}/{cos^2\ x} dx=1/{cos\ x}+c=sec\ x+c$
  19. $∫{sin^2\ x}/{cos\ x} dx=ln|tan(π/4+x/2)|-sin\ x+c$
  20. $∫tan^2\ x\ dx=tan\ x-x+c$
  21. $∫cot\ x\ dx=ln|sin\ x|+c$
  22. $∫{cos\ x}/{sin^2\ x}\ dx=-1/{sin\ x}+c=-cosec\ x+c$
  23. $∫{cos^2\ x}/{sin\ x}\ x\ dx=ln|tan(x/2)|+cos\ x+c$
  24. $∫cot^2\ x\ dx=-cot\ x-x+c$
  25. $∫{dx}/{sin\ x\ cos\ x}=ln|tan\ x|+c$
  26. $∫{dx}/{sin^2\ x\ cos\ x}=-1/{sin\ x}+ln|tan(π/4+x/2)|+c$
  27. $∫{dx}/{sin\ x\ cos^2\ x}=1/{cos\ x}+ln|tan(x/2)|+c$
  28. $∫{dx}/{sin^2\ x\ cos^2\ x}=tan\ x-cot\ x+c$
  29. $∫sin\ mx\ sin\ nx\ dx=-{sin(m+n)x}/{2(m+n)}+{sin(m-n)x}/{2(m-n)}+c, m^2≠n^2$
  30. $∫sin\ mx\ cos\ nx\ dx=-{cos(m+n)x}/{2(m+n)}-{cos(m-n)x}/{2(m-n)}+c, m^2≠n^2$
  31. $∫cos\ mx\ cos\ nx\ dx={sin(m+n)x}/{2(m+n)}+{sin(m-n)x}/{2(m-n)}+c, m^2≠n^2$
  32. $∫sec\ x\ tan\ x\ dx=sec\ x+c$
  33. $∫cosec\ x\ cot\ x\ dx=-cosec\ x+c$
  34. $∫sin\ x\ cos^n\ x\ dx=-{cos^{n+1}\ x}/{n+1}+c$
  35. $∫sin^n\ x\ cos\ x\ dx={sin^{n+1}\ x}/{n+1}+c$
  36. $∫sin^{-1}\ x\ dx=x\ sin^{-1}\ x+√{1-x^2}+c$
  37. $∫cos^{-1}\ x\ dx=x\ cos^{-1}\ x-√{1-x^2}+c$
  38. $∫tan^{-1}\ x\ dx=x\ tan^{-1}\ x-1/2\ ln(x^2+1)+c$
  39. $∫cot^{-1}\ x\ dx=x\ cot^{-1}\ x+1/2\ ln(x^2+1)+c$

Post Tags: Integration


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