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How do you integrate (a^x + b^x)^3 /((a^x)*(b^x))

  • #1

Homework Statement



Integrate (a^x + b^x)^3 /((a^x)*(b^x))


The Attempt at a Solution



I have only 2 semesters of calculus under my belt yet nothing in my experience has taught me to do anything like this
 

Answers and Replies

  • #2
tiny-tim
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Homework Helper
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Hi The_ArtofScience! :smile:

(try using the X2 tag just above the Reply box :wink:)
Integrate (a^x + b^x)^3 /((a^x)*(b^x))
erm :redface: … just expand it …

(a2/b)x + … :smile:
 
  • #3
The algebra that I did from expanding it like you suggested was:

(a^3x + 3a^2x b^x + 3a^x b^2x + b^3x)/(a^x b^x)
= (a^2 /b)^x + 3a^x + 3b^x + (b^2/a)^x

But then what do I do with this strange exponent? I don't know how to integrate an expression like b^x
 
  • #4
Born2bwire
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You can find it easily enough by a search but you can derive it pretty easily too.

[tex]\log_b(b^x) = x[/tex]
[tex]\frac{d}{dx}\log_b(b^x) = 1[/tex]
[tex]\frac{d}{dx}\log_b(b^x) = \frac{d}{du} \log_b(u) \cdot \frac{d b^x}{dx}[/tex] where [tex]u = b^x[/tex].

Convert [tex]log_b(u) = \frac{\ln(u)}{\ln(b)}[/tex] and go from there.
 
  • #5
tiny-tim
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Homework Helper
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But then what do I do with this strange exponent? I don't know how to integrate an expression like b^x
(please use the X2 tag just above the Reply box … it's much easier to read)

Yes, use Born2bwire's :smile: method,

or write bx = (eln(b))x = eln(b)x,

so (bx)' = ln(b)bx :wink:
 
  • #6
Born2bwire
Science Advisor
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Oh sure, do it the easy and efficient way. ;)
 
  • #7
Hi I just read the messages and I appreciate both of your help. Thanks! I get it now. It was silly for me not to see it before I asked
 
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