Integration using Substitution: Fresh Take

  • Identify when to use substitution to simplify and solve integrals
  • Apply substitution methods to find indefinite integrals
  • Apply substitution methods to find definite integrals

Substitution for Indefinite Integrals

The Main Idea 

  • Integration by substitution is a technique for evaluating integrals. It’s useful when the integrand is the result of a chain-rule derivative
  • The method involves changing variables to simplify the integral
  • Key form to recognize:
    • [latex]\int f(g(x))g'(x)dx[/latex]
  • Substitution Theorem:
    • For [latex]u = g(x)[/latex] where [latex]g'(x)[/latex] is continuous: [latex]\int f(g(x))g'(x)dx = \int f(u)du = F(u) + C = F(g(x)) + C[/latex]
  • Process:
    • Choose [latex]u = g(x)[/latex] such that [latex]g'(x)[/latex] is part of the integrand
    • Express the integral in terms of [latex]u[/latex]
    • Integrate with respect to [latex]u[/latex]
    • Substitute back to express the result in terms of [latex]x[/latex]
  • Sometimes need to adjust by a constant factor when [latex]du[/latex] doesn’t match exactly
  • May need to express [latex]x[/latex] in terms of [latex]u[/latex] to eliminate all [latex]x[/latex] terms

Use substitution to find the antiderivative of [latex]\displaystyle\int 3{x}^{2}{({x}^{3}-3)}^{2}dx.[/latex]

Use substitution to find the antiderivative of [latex]\displaystyle\int {x}^{2}{({x}^{3}+5)}^{9}dx.[/latex]

Use substitution to evaluate the integral [latex]\displaystyle\int \frac{ \cos t}{{ \sin }^{2}t}dt.[/latex]

Use substitution to evaluate the indefinite integral [latex]\displaystyle\int { \cos }^{3}t \sin tdt.[/latex]

Substitution for Definite Integrals

The Main Idea 

  • Substitution can be applied to definite integrals
  • The limits of integration must be adjusted when changing variables
  • The technique combines substitution with the Fundamental Theorem of Calculus
  • Substitution Theorem for Definite Integrals:
    • If [latex]u = g(x)[/latex] and [latex]g'(x)[/latex] is continuous on [latex][a,b][/latex], then: [latex]\int_a^b f(g(x))g'(x)dx = \int_{g(a)}^{g(b)} f(u)du[/latex]
  • Process:
    • Choose [latex]u = g(x)[/latex] as in indefinite integration
    • Express the integrand in terms of [latex]u[/latex]
    • Change the limits of integration from [latex]x[/latex] to [latex]u[/latex] values
    • Integrate with respect to [latex]u[/latex]
    • Evaluate the integral using the new limits
  • Alternative Approach:
    • Perform substitution without changing limits
    • Find the antiderivative in terms of [latex]u[/latex]
    • Substitute back to [latex]x[/latex] before evaluating at the original limits
  • Combining Techniques:
    • Substitution may be used alongside other integration techniques
    • Trigonometric identities might be needed before substitution

Use substitution to evaluate the definite integral [latex]{\displaystyle\int }_{-1}^{0}y{(2{y}^{2}-3)}^{5}dy.[/latex]

Use substitution to evaluate [latex]{\displaystyle\int }_{0}^{1}{x}^{2} \cos \left(\frac{\pi }{2}{x}^{3}\right)dx.[/latex]