List of basic formulas and steps associated with the method of substitution of variable in an indefinite integral:

  1. The general form of variable substitution is:
    If ( u = g(x) ), then ( du = g'(x) \, dx ).
  2. Indefinite integral:
    [
    \int f(g(x)) g'(x) \, dx = \int f(u) \, du
    ]
  3. Back substitution:
    After integrating, don’t forget to return the variable to its original form:
    [
    \int f(u) \, du = F(u) + C \implies F(g(x)) + C
    ]
  4. Example:
    For the integral ( \int 2x \cos(x^2) \, dx ):
  • We replace the variable: (u = x^2), then (du = 2x \, dx).
  • The integral becomes: ( \int \cos(u) \, du = \sin(u) + C ).
  • We return to the original variable: ( \sin(x^2) + C ).

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