Lecture Notes: Integration by Parts

Introduction to Integration Techniques

• Complex integrals require special techniques.
• Many integration rules mirror differentiation rules:
  • Substitution Rule: Similar to chain rule in differentiation.
  • Integration by Parts: Corresponds to the product rule in differentiation.

Product Rule Recap

• Derivative of a product: [ (f \cdot g)' = f g' + f' g ]
• Integration can "reverse" differentiation.
• Integral of the derivative equals the product of two functions.

Deriving the Integration by Parts Formula

• Formula: [ \int (f g') = fg - \int (f'g) ]
• Useful for integrating a product of two functions.

Simplified Notation

• Represent:
  • ( f(x) ) as ( u )
  • ( g'(x) ) as ( dv )
• Formula becomes: [ \int u\ dv = uv - \int v\ du ]

Strategy for Using Integration by Parts

• Select ( f(x) ) such that ( f'(x) ) is simpler to integrate.
• Evaluate product integrals by simplifying components.

Example 1: Integrate ( x \sin x \ dx )

• Assign ( f(x) = x ), ( g'(x) = \sin x )
• Calculate:
  • ( \int x \sin x = -x \cos x + \int \cos x )
  • Result: (-x \cos x + \sin x + C)

Example 2: Integrate ( \ln x \ dx )

• Assign ( u = \ln x ), ( dv = dx )
• Calculate:
  • ( \int \ln x = x \ln x - \int x^{-1} )
  • Result: ( x \ln x - x + C )

Example 3: Integrate ( \frac{\ln x}{x^2} \ dx )

• Assign ( u = \ln x ), ( dv = \frac{dx}{x^2} )
• Calculate:
  • ( \int \frac{\ln x}{x^2} = -\frac{\ln x}{x} - \int x^{-2} )
  • Result: (-\frac{\ln x}{x} - \frac{1}{x} + C)

Example 4: Integrate ( (\ln x)^2 \ dx )

• Assign ( u = (\ln x)^2 ), ( dv = dx )
• Recursive integration by parts:
  • ( \int (\ln x)^2 = x(\ln x)^2 - 2 \int \ln x )
  • Result: ( x(\ln x)^2 - 2(x \ln x - x) + C)

Conclusion

• Integration by parts helps when integrand functions aren't directly related but one simplifies upon differentiation.
• Choose components wisely to simplify further integrations.

Next Steps

• One more integration trick to learn.
• Check comprehension and practice more examples.