Integration by Parts 5x ln(4x)dx

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SUMMARY

The integral ∫5x ln(4x)dx can be evaluated using integration by parts, specifically applying the formula ∫udv = uv - ∫vdu. The discussion highlights the importance of correctly identifying u and dv; in this case, u is set to ln(4x). A common mistake noted is the derivative of ln(4x), which simplifies to 1/x, as ln(4x) can be expressed as ln(x) + ln(4). This clarification led to the correct evaluation of the integral.

PREREQUISITES
  • Understanding of integration by parts
  • Knowledge of logarithmic differentiation
  • Familiarity with the chain rule in calculus
  • Basic skills in evaluating definite and indefinite integrals
NEXT STEPS
  • Practice more problems using integration by parts
  • Study the properties of logarithmic functions
  • Learn about the chain rule in greater detail
  • Explore advanced integration techniques, such as integration by substitution
USEFUL FOR

Students learning calculus, particularly those focusing on integration techniques, as well as educators seeking to clarify integration by parts and logarithmic differentiation.

sashab
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Homework Statement



Use integration by parts to evaluate the integral.
∫5x ln(4x)dx


Homework Equations



∫udv = uv - ∫vdu

The Attempt at a Solution


So here's my solution:
tumblr_n1a0635Kjb1tsd2vco1_500.jpg


But the computer is telling me I'm wrong :( We haven't learned how to integrate lnx yet, so the only choice I have is to make u = ln(4x) (even our textbook does this). Any help would be really appreciated! Thanks :)
 
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sashab said:

Homework Statement



Use integration by parts to evaluate the integral.
∫5x ln(4x)dx


Homework Equations



∫udv = uv - ∫vdu

The Attempt at a Solution


So here's my solution:
tumblr_n1a0635Kjb1tsd2vco1_500.jpg


But the computer is telling me I'm wrong :( We haven't learned how to integrate lnx yet, so the only choice I have is to make u = ln(4x) (even our textbook does this). Any help would be really appreciated! Thanks :)
What is the derivative of ln(4x) ?
 
SammyS said:
What is the derivative of ln(4x) ?

Oh whoops! I can't believe I didn't notice such an obvious mistake. Thanks, I got the right answer now. :)
 
By the chain rule, the derivative of ln(4x) is (1/4x) times the derivative of 4x so (1/4x)(4)= 1/x.

Even simpler: ln(4x)= ln(x)+ ln(4) so its derivative is 1/x.
 

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