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

The_ArtofScience · May 20, 2009

1. The problem statement, all variables and given/known data

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

3. 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

tiny-tim · May 20, 2009

Hi The_ArtofScience!

(try using the X² tag just above the Reply box )

The_ArtofScience said: ↑

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

erm … just expand it …

(a²/b)^x + …

The_ArtofScience · May 21, 2009

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

Born2bwire · May 21, 2009

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.

tiny-tim · May 21, 2009

The_ArtofScience said: ↑

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 method,

or write b^x = (e^ln(b))x = e^ln(b)x,

so (b^x)' = ln(b)b^x

Born2bwire · May 21, 2009

Oh sure, do it the easy and efficient way. ;)

The_ArtofScience · May 22, 2009

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

Log in or Sign up

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

The_ArtofScience

tiny-tim

The_ArtofScience

Born2bwire

tiny-tim

Born2bwire

The_ArtofScience

Prove that the integr of x² = (1/3)(b³-a³) (Replies: 4)

Sets: Is A ⊆ B? A={x ∈ ℤ | x ≡ 7 (mod 8)} B={x ∈ ℤ | x ≡ 3 (mod 4)} (Replies: 6)

Integral of x / ((a-x)^2+b^2)^2 ? (Replies: 1)

How to integrate x^(-a)*e^(-b/x), where a, b are constants? (Replies: 4)

Integrating dx/dt=k(a-x)(b-x) (Replies: 13)