Integration by Parts

Integration by Parts Method for Indefinite Integrals:

  • Principle: Used for integrating the product of two functions.
  • Formula: ∫ u dv = uv - ∫ v du, where u and v are functions of x.
  • Derivation: Derived from the product rule of differentiation (d(uv)/dx = u dv/dx + v du/dx).

Steps for Integration by Parts:

  1. Choose u and dv:

    • u: Select a function that can be easily differentiated.
    • dv: Choose a function whose integral can be easily computed.
  2. Differentiate and Integrate:

    • Calculate du and v by differentiating u and integrating dv.
  3. Apply the Formula:

    • Substitute the values of u, du, v, and dv into the integration by parts formula.
  4. Solve for the Integral:

    • Evaluate the resulting integral, which may involve repeating the integration by parts method or simplification.

Example:

Let's solve the indefinite integral ∫ x * ex dx using the integration by parts method:

Steps:

  1. Choose u and dv:

    • Let u=x (easily differentiable)
    • dv=exdx (integral of ex is ex)
  2. Differentiate and Integrate:

    • Calculate du=dx and v=ex
  3. Apply the Formula:

    • Substitute into the integration by parts formula:

    xexdx=xexexdx

  4. Solve for the Integral:

    • The integral of ex is ex, so the solution is:

    xexex+C

Tips for Integration by Parts:

  • Choose u and dv Wisely: Select u and dv in such a way that du or v can be easily computed.
  • Usefulness in Repeated Integration: Sometimes, repeated application of integration by parts might be necessary to solve the integral completely.
  • Identify Patterns: Look for patterns that might simplify the integral or make it easier to solve.