Simplifying Integration: Strategies and Tips for Tricky Expressions

[catalyst55]

hey,

im trying to integrate the following:

e^(2x)sqrt(e^x - 1)

if only it were e^x at the front - it'd be so much easier!

anyway, letting u=e^x-1 and doing everything i eventually get the following (and this is where i get stuck):

[ u^(5/2)+2u^(3/2)+u^(1/2) ] [ 1/(u+1) ] du

this is correct because, according to the computer, it simplifies to u^(3/2) + u^(1/2), which then gives the right answer.

so my question, really, is: how do we simplify [ u^(5/2)+2u^(3/2)+u^(1/2) ] [ 1/(u+1) ] ??


or am i approaching this the wrong way (which i suspect because it's a multiple choice question)??

cheers

 

[J77]

Looks like a nice substitution...

after which I end up with:

$$\int e^{2x}\sqrt{u}\frac{du}{e^x}=\int e^{x}\sqrt{u} du=\int (u+1)\sqrt{u} du$$

which is what your computer says.

(for your simplification: take a factor of $$u^{1/2}$$ out and factorise...)

 

[neutrino]

$$(\sqrt u)^5 + (\sqrt u)^3 + \sqrt u= \sqrt u \left(u^2 + 2u + 1)\right$$

 

[nrqed]

hey,

im trying to integrate the following:

e^(2x)sqrt(e^x - 1)

if only it were e^x at the front - it'd be so much easier!

anyway, letting u=e^x-1 and doing everything i eventually get the following (and this is where i get stuck):

[ u^(5/2)+2u^(3/2)+u^(1/2) ] [ 1/(u+1) ] du

this is correct because, according to the computer, it simplifies to u^(3/2) + u^(1/2), which then gives the right answer.

so my question, really, is: how do we simplify [ u^(5/2)+2u^(3/2)+u^(1/2) ] [ 1/(u+1) ] ??

$(u^{5/2} + 2 u^{3/2} + u^{1/2}) = u^{1/2} ( u^2+ 2 u + 1) = u^{1/2} (u+1)^2$. That's all there was to it!

Patrick

 

[StatusX]

As that's sort of a lucky break peculiar to this example, a better substitution for this general type of problem might be $u=\sqrt{e^x-1}$. Then $e^x=u^2+1$ and $e^x dx = 2u du$.

 

[catalyst55]

oh, thanks a lot guys.

it's so much quicker and easier if i just cancel (u+1)^2 and 1/(u+1), leaving just (u+1) instead of expanding and then trying to simplify...

seems like i approached this q in the most inefficient of ways -- i even had x= ln (u+1) lol... i had like e^[ln(u+1)] = u+1... don't know what i was thinking...

oh, by the way, do any of you guys know where i could find a good summary of all of the different types of antidifferation (ie like a compendium with an example of every particular type of question or someting)?

cheers

 

[J77]

You'll usually find "antidifferentiation" in the front of most calculus books - inside covers.

Or Abramowitz and Stegun: Handbook of Mathematical Functions is the Giant Haystacks of the formulae world.