Integration by Parts Lecture Notes

Overview

• Formula for integration by parts: ( \int u , dv = uv - \int v , du )
• Used when integration is complex and can be simplified via differentiation and integration of parts of the function.
• Often requires the method to be applied multiple times, similar to L'Hôpital's Rule.

Example 1: ( \int 7x , e^{8x} , dx )

• Choosing ( u ) and ( dv ):
  • ( u = 7x ), ( dv = e^{8x} , dx )
  • ( du = 7 , dx )
  • ( v = \frac{1}{8} e^{8x} ) (using chain rule)
• Integration by Parts:
  • ( uv = \frac{7}{8} x e^{8x} )
  • ( \int v , du = \int \frac{7}{8} e^{8x} , dx )
  • Result: ( \frac{7}{8} x e^{8x} - \frac{7}{64} e^{8x} + \frac{7}{64} )

Example 2: ( \int_0^{\pi} x^2 \sin(2x) , dx )

• Choosing ( u ) and ( dv ):
  • ( u = x^2 ), ( dv = \sin(2x) , dx )
  • ( du = 2x , dx )
  • ( v = -\frac{1}{2} \cos(2x) )
• First Integration by Parts:
  • ( u v = -\frac{1}{2} x^2 \cos(2x) )
  • ( \int v , du = \int x \cos(2x) , dx )
• Second Integration by Parts on ( \int x \cos(2x) , dx ):
  • New ( u = x ), ( dv = \cos(2x) , dx )
  • ( du = dx ), ( v = \frac{1}{2} \sin(2x) )
  • Result: ( -\frac{1}{2} x^2 \cos(2x) + x \sin(2x) - \frac{1}{4} \cos(2x) )
• Final Result:
  • Evaluate from 0 to ( \pi ), resulting in ( -\frac{\pi^2}{2} )

Tabular Method

• Shortcut for repeated integration by parts when function reduces to constant.
• Create columns for derivatives of ( u ) and antiderivatives of ( dv ) with alternating signs.
• Multiply across rows and sum, considering alternating signs.

Example 3: ( \int \ln(s) , ds )

• Choosing ( u ) and ( dv ):
  • ( u = \ln(s) ), ( dv = ds )
  • ( du = \frac{1}{s} , ds ), ( v = s )
• Integration by Parts:
  • ( uv = s \ln(s) )
  • ( \int v , du = \int 1 , ds = s )
  • Result: ( s \ln(s) - s + C )

General Tips

• Choose ( u ) such that its derivative is simpler than ( u ).
• Check integrability of ( \int v , du ) after substitution.
• Use tabular method for repeated integration by parts with polynomial functions.