Evaluating Integral with a natural log

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SUMMARY

The discussion focuses on evaluating the integral of the function 9^s with respect to s. The user correctly identifies the integration by parts method, setting u = 9s and dv = 9^s ds. The solution progresses to v = 3^(2s)/(2ln(3)), but it is confirmed that the integral can also be expressed as 9^s/ln(9). Both approaches yield valid results for the integral.

PREREQUISITES
  • Understanding of integral calculus, specifically integration by parts.
  • Familiarity with exponential functions and their properties.
  • Knowledge of logarithmic identities, particularly natural logarithms.
  • Ability to manipulate and simplify expressions involving exponents.
NEXT STEPS
  • Study the method of integration by parts in detail.
  • Explore the properties of exponential functions and their integrals.
  • Learn about logarithmic differentiation and its applications.
  • Practice solving integrals involving different bases and logarithmic forms.
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Students studying calculus, mathematics educators, and anyone looking to deepen their understanding of integral evaluation techniques.

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Homework Statement


[tex]\int9s9^s ds[/tex]

Homework Equations


∫udv=uv-∫9^sds-∫vdu

The Attempt at a Solution


u=9s
du=9ds

dv=9^s ds
v=∫9^sds
=∫3^(2s) ds
=3^(2s)/[2ln(3)]
this is as far as i have gotten. Am i correct so far?
 
Last edited:
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Yes, that is correct. You didn't really need to reduce to 3, that is exactly the same as
[tex]\int 9^s ds= 9^s/ln(9)[/tex]
 

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