Section 5.2: Integration by Substitution

General Indefinite Integral Formulas

  1. ∫ [f(x)]nf'(x) dx = [f(x)]n+1/(n + 1) + C, n ≠ -1
  2. ∫ ef(x)f'(x) dx = ef(x) + C
  3. ∫ f'(x)/f(x) dx = ln|f(x)| + C
  4. ∫ un du = un+1/(n + 1) + C, n ≠ -1
  5. ∫ eu du = eu + C
  6. ∫ 1/u du = ln|u| + C

Differentials

If y = f(x) defines a differentiable function, then:

  1. The differential dx of the independent variable x is an arbitrary real number.
  2. The differential dy of the dependent variable y is defined as the product of f'(x) and dx:
    dy = f'(x) dx

Integration by Substitution

  1. Select a substitution that appears to simplify the integrand. In particular, try to select u so that du is a factor in the integrand.
  2. Express the integrand entirely in terms of u and du, completely eliminating the original variable and its differential.
  3. Evaluate the new integral if possible.
  4. Express the antiderivative found in step 3 in terms of the original variable.