不定積分の公式

目次


積分の定義

基本性質

部分積分

置換積分

不定積分の公式


テスト

Logをみる

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  1. ∫adx = ax+C (a は定数)
  2. ∫x^adx = x^(a+1)/(a+1)+C (a != -1)
  3. ∫x^(-1)dx = log(|x|)+C
  4. ∫1/sqr(1-x^2)dx = arcsin(x)+C
  5. ∫-1/sqr(1-x^2)dx = arccos(x)+C
  6. ∫1/(1+x^2)dx = arctan(x)+C
  7. ∫exp(x)dx = exp(x)+C
  8. ∫a^xdx = a^x/log(a)+C
  9. ∫log(x)dx = x(log(x)-1)+C
  10. ∫sin(x)dx = -cos(x)+C
  11. ∫cos(x)dx = sin(x)+C
  12. ∫tan(x)dx = -loge|cos(x)|+C = loge|sec(x)|+C
  13. ∫cot(x)dx = loge|sin(x)|+C
  14. ∫sec(x)dx = loge|sec(x)+tan(x)|+C
  15. ∫sin2(x)dx
    = 1 (x-sin(x)cos(x))+C
    2
    = 1 x - 1 sin(2x)+C
    2 4
  16. ∫cos2(x)dx
    = 1 (x+sin(x)cos(x))+C
    2
    = 1 x + 1 sin(2x)+C
    2 4
  17. ∫tan2(x)dx = tan(x)-x+C
  18. ∫cot2(x)dx = -cot(x)-x+C
  19. ∫sin(ax)sin(bx)dx
    = sin((a-b)x) - sin((a+b)x) +C
    2(a-b) 2(a+b)
  20. ∫sin(ax)cos(bx)dx
    = - cos((a-b)x) - cos((a+b)x) +C
    2(a-b) 2(a+b)
  21. ∫cos(ax)cos(bx)dx
    = sin((a-b)x) + sin((a+b)x) +C
    2(a-b) 2(a+b)
  22. ∫xsin(x)dx = sin(x)-xcos(x)+C
  23. ∫xcos(x)dx = cos(x)+xsin(x)+C
  24. ∫x2sin(x)dx = (2-x2)cos(x)+2xsin(x)+C
  25. ∫x2cos(x)dx = (x2-2)sin(x)+2xcos(x)+C
  26. ∫(1/cos2(x))dx = ∫(sec2(x))dx = tan(x)+C
  27. ∫arcsin(x)dx = 1/2 log((1-cos(x))/(1+cos(x))) + C
    = log(|tan(x/2)|) + C
  28. ∫arccos(x)dx = 1/2 log((1+sin(x))/(1-sin(x))) + C
    = log(|(1+tan(x/2))/(1-tan(x/2))|) + C
  29. ∫arctan(x)dx = log(|sin(x)|) + C