1.5 Substitution

The substitution method is a powerful tool for simplifying complex integrals. By changing variables, we can transform tricky integrands into more manageable forms. This technique is especially handy when dealing with composite functions or expressions that resemble the chain rule.

Substitution works for both indefinite and definite integrals, with slight variations in the process. Recognizing when to use substitution is key - look for composite functions or inner functions paired with their derivatives. It's a versatile method that often paves the way for solving challenging integration problems.

Substitution Method for Integration

Substitution for indefinite integrals

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• Simplifies and evaluates integrals by changing the variable of integration
• Particularly useful for integrands that are composite functions (function inside another function)
• Steps to apply substitution:
  1. Identify part of integrand to replace with new variable $u$
  2. Determine differential of $u$ ($du$) by differentiating $u$ with respect to original variable
  3. Replace identified part of integrand with $u$ and corresponding differential with $du$
  4. Simplify new integrand and evaluate integral with respect to $u$
  5. Substitute back original expression for $u$ to obtain antiderivative in terms of original variable
• Substitution is closely related to the chain rule for derivatives, but applied in reverse

Substitution in definite integrals

• Substitution also applies to definite integrals
• Additional steps when using substitution in definite integrals:
  • Change limits of integration to correspond to new variable $u$
    • Evaluate substitution $u$ at original lower and upper limits to find new limits
  • Evaluate definite integral with respect to $u$ using new limits
  • Substitute back original expression for $u$ to obtain final result
• The fundamental theorem of calculus is applied after substitution to evaluate the definite integral

Recognition of substitution-suitable integrands

• Characteristics of integrands suitable for substitution:
  • Integrand is a composite function
  • Inner function appears along with its derivative multiplied by some other term
• Common substitutions:
  • Trigonometric functions ($u = \sin x$, $u = \cos x$, $u = \tan x$)
  • Exponential functions ($u = e^x$, $u = a^x$)
  • Logarithmic functions ($u = \ln x$, $u = \log_a x$)
  • Square roots ($u = \sqrt{x}$, $u = \sqrt{ax + b}$)
  • Polynomials ($u = ax + b$, $u = ax^2 + bx + c$)
• Choice of substitution depends on form of integrand and aim to simplify integral
• Substitution should result in simpler integrand that can be easily integrated using known techniques

Advanced Integration Techniques

• Integration by parts: Used when the integrand is a product of functions
• Inverse functions: Substitution can be particularly useful when dealing with integrals involving inverse functions
• Differential notation: The use of $du$ in substitution is an example of differential notation, which helps visualize the substitution process
• Change of variables: A more general form of substitution used in multivariable calculus