Integration by Parts: Understanding dv & dx

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SUMMARY

The discussion focuses on the integration technique known as "Integration by Parts," specifically addressing the necessity of including dx in the differential dv. It establishes that dv must always include dx because a differential represents a change in a variable, and cannot equate to an ordinary function. The relationship is rooted in the fundamental principles of calculus, particularly the product rule for derivatives, which is expressed as d(uv) = udv + vdu.

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  • Understanding of calculus fundamentals, specifically differentiation and integration.
  • Familiarity with the product rule for derivatives.
  • Knowledge of differential notation and its significance in calculus.
  • Basic grasp of the concept of differentials in mathematical analysis.
NEXT STEPS
  • Study the derivation and applications of the product rule in calculus.
  • Explore advanced integration techniques, including trigonometric and logarithmic integrals.
  • Learn about the Fundamental Theorem of Calculus and its implications for integration.
  • Investigate the concept of differentials in more depth, including their role in differential equations.
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Students of calculus, mathematics educators, and anyone seeking a deeper understanding of integration techniques and the underlying principles of differentials.

Bashyboy
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I understand this integration technique, for the most part. One thing I am curious to know is why, when you do your rudimentary substitution for this particular technique, does dv have to always include dx?
 
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"Integration by parts" is just the integral version of the "product rule" for derivatives:
d(uv)= udv+ vdu. We can write that as udv= d(uv)- vdu and integrate both sides:
[itex]\int udv= \int d(uv)- \int vdu[/itex]. Of course, [itex]\int d(uv)= uv[/itex].

As for "does dv have to always include dx?", yes, of course. "dv" is a differential and you cannot have an "ordinary" function equal to a differential. A differential can only be equal to another differential.
 
HallsofIvy said:
As for "does dv have to always include dx?", yes, of course. "dv" is a differential and you cannot have an "ordinary" function equal to a differential. A differential can only be equal to another differential.

Is there a reason for this? Or have mathematicians defined this to be true?
 

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