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Indefinite Integral Formulas


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    Rules

  1. Multiplication by constant

    k f(x)  dx = k f(x)  dx
  2. Addition/Subtraction

    [ f(x) ± g(x) ]  dx = f(x)  dx ± g(x)  dx
  3. Integration by parts

    f ( x ) g ( x )   d x = f ( x ) g ( x ) f ( x ) g ( x )   d x
  4. Integration by substitution

    i.   u=u(x) , so,  du=u(x) dx
    ii.   x=x(u) , so,  dx=x(u) du

    Using i and/or ii, solve the integral and then substitute back

    f ( x )   d x = f ( x ( u ) )   x ( u )   d u
    or f(x) dx=f(x(u)) 1u(x)du
  5. Basic Indefinite Integrals

  6. k dx = kx +c
  7. x n dx = x n + 1 n + 1 +c , n 1
  8. Trignometric Indefinite Integrals

  9. sin(ax)  dx = cos(ax) a +c
  10. cos(ax)  dx = sin(ax) a +c
  11. sin 2 (ax)  dx = x 2 sin(2ax) 4a +c
  12. cos 2 (ax)  dx = x 2 + sin(2ax) 4a +c
  13. sec 2 x  dx = tan x +c
  14. csc 2 x  dx = cot x +c
  15. sec x   tan x  dx = sec x +c
  16. csc x   cot x  dx = csc x +c
  17. x   sin(ax)  dx = sin(ax) ax   cos(ax) a 2 +c
  18. x   cos(ax)  dx = cos ( a x ) + a x   sin ( a x ) a 2 +c
  19. tan x  dx = ln | cos x | +c
  20. cot x  dx = ln | sin x | +c
  21. sec x  dx = ln | sec x + tan x | +c
  22. csc x  dx = ln | csc x + cot x | +c
  23. Exponential Indefinite Integrals

  24. e a x dx = e a x a +c
  25. a x dx = a x ln ( a ) +c , a>0  &  a1
  26. ln x dx = x ln(x) x +c
  27. xm eax dx = xm eax a ma xm1 eax dx
  28. xeax dx = x eax a eax a2 = ( ax1 )   eax a2
  29. Fractional Indefinite Integrals

  30. 1 x  dx = ln | x | +c
  31. 1 a 2 x 2  dx = sin 1 ( x a ) +c
  32. 1 a 2 + x 2  dx = tan 1 ( x a ) a +c
  33. 1 x   x 2 a 2  dx = sec 1 ( x a ) a +c
 
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