Substitution Method

Substitution Method for Indefinite Integrals:

  • Principle: Involves substituting variables to simplify complex integrals.
  • Steps:
    1. Choose Substitution: Select a suitable substitution involving a new variable, often denoted by u.
    2. Compute Differential: Find du = f'(x) dx, where f'(x) represents the derivative of the chosen substitution variable u with respect to x.
    3. Substitute: Express the integral in terms of the new variable u and du.
    4. Integrate: Solve the integral in terms of u.
    5. Resubstitute: Convert the solution back to the original variable x.

Example:

Let's solve the indefinite integral ∫ 2x(x2+1)5 dx using the substitution method:

Steps:

  1. Choose Substitution: Let u=x2+1.

  2. Compute Differential: Find du=2xdx by differentiating u with respect to x.

  3. Substitute: Rewrite the integral in terms of u and du:

    2x(x2+1)5dx=u5du

  4. Integrate: Now, integrate the expression in terms of u:

    u5du=u66+C

  5. Resubstitute: Replace u with the original substitution:

    (x2+1)66+C

Tips for Substitution Method:

  • Appropriate Choices: Select substitutions that simplify the integral or reduce it to a standard form.
  • Derivative Alignment: Ensure the presence of the differential du in the integral to match with the derivative of the substituted variable.
  • Back-substitution: Always express the final result in terms of the original variable.