THE SUBSTITUTION RULE

5 Integrals

5.5 The Substitution Rule

Indefinite Integrals

  • If u = g(x) is differentiable and f is continuous on the interval I, apply:
    \int f(g(x)) g'(x) \, dx = \int f(u) \, du

  • Remember the Substitution Rule is proved using the Chain Rule for differentiation.

Definite Integrals

  • For continuous functions over [a, b]:
    \int{a}^{b} f(g(x)) g'(x) \, dx = \int{g(a)}^{g(b)} f(u) \, du

Examples:

Example 1: Find ( \int (4x^3 + 5) \cos(2x) \, dx )

  • Use substitution ( u = 4x^3 + 5 )

  • Compute ( du = 12x^2 dx )

  • Transform integral using Substitution Rule to evaluate.

Example 7: Evaluate ( \int_0^4 (2x + 1) \sin(6) \, dx )

  • Use substitution ( u = 2x + 1 ) ( \Rightarrow ) Limits: When x=0, u=1; When x=4, u=9.

  • Don't return to variable x after integrating; evaluate in terms of u between the new limits.