Lecture on Integration by Parts

Key Concepts

• Integration by Parts Formula:

  • Based on the product rule for differentiation:
  • [ \int u , dv = uv - \int v , du ]
  • Choose ( u ) and ( dv ) from the integrand such that differentiating ( u ) and integrating ( dv ) simplifies the problem.

• LIATE Rule:

  • A useful guideline for selecting ( u ):
    • Logarithmic functions (( L ))
    • Inverse trigonometric functions (( I ))
    • Algebraic functions (( A ))
    • Trigonometric functions (( T ))
    • Exponential functions (( E ))
  • Priority decreases from top to bottom.

Examples

• Example 1: Solving ( \int x \cos(x) , dx )

  • Choose ( u = x ), ( dv = \cos(x) , dx )
  • Differentiate ( u ): ( du = dx )
  • Integrate ( dv ): ( v = \sin(x) )
  • Apply formula: ( \int x \cos(x) , dx = x \sin(x) - \int \sin(x) , dx = x \sin(x) + \cos(x) + C )

• Example 2: Solving ( \int \ln(x) , dx )

  • Choose ( u = \ln(x) ), ( dv = dx )
  • Differentiate ( u ): ( du = \frac{1}{x} , dx )
  • Integrate ( dv ): ( v = x )
  • Apply formula: ( \int \ln(x) , dx = x \ln(x) - \int x \cdot \frac{1}{x} , dx = x \ln(x) - \int 1 , dx = x \ln(x) - x + C )

Applications

• Used when the standard integration techniques (e.g., substitution) are not applicable.
• Commonly used in solving integrals involving products of polynomial and trigonometric functions, logarithmic functions, and exponential functions.

Practice Problems

1. ( \int x \sin(x) , dx )
2. ( \int e^x \cos(x) , dx )
3. ( \int x^2 e^x , dx )
4. ( \int x \ln(x) , dx )

Tips

• Carefully choose ( u ) and ( dv ) for simplicity.
• The LIATE rule is not absolute; sometimes experimentation is necessary.
• Check your work by differentiating the result to see if it matches the original integrand.