Integration by Parts 5x ln(4x)dx

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Homework Help Overview

The problem involves evaluating the integral ∫5x ln(4x)dx using integration by parts. The subject area pertains to calculus, specifically integration techniques.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to apply integration by parts but expresses confusion regarding the integration of ln(4x). They consider using u = ln(4x) as suggested by their textbook. Another participant questions the derivative of ln(4x), which leads to a realization of a mistake in the original poster's approach.

Discussion Status

The discussion has progressed with participants exploring the derivative of ln(4x) and clarifying the original poster's misunderstanding. A participant has acknowledged a mistake and indicated they have arrived at the correct answer, although the overall consensus on the method remains implicit.

Contextual Notes

The original poster mentions that they have not yet learned how to integrate ln(x), which may limit their approach to the problem.

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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