List of basic formulas and steps associated with the method of substitution of variable in an indefinite integral:
- The general form of variable substitution is:
If ( u = g(x) ), then ( du = g'(x) \, dx ). - Indefinite integral:
[
\int f(g(x)) g'(x) \, dx = \int f(u) \, du
] - 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
] - 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 ).