How to Integrate ∫(e^2lnx)(10x)dx?

[Wonderland]

Hi everyone,

I just started to study Physics and I have a very simple question about integration. I'm blocked on this for a while. I'm trying to solve a differential equation for which I need to integrate the expression:
∫(e^2lnx)(10x)dx

Does someone can tell me how to do it? I can't find this exact expression in formulas, and I've tried substitution with u = e^2lnx or with u = 2lnx and I've also tried to use ∫u'v = uv - ∫ uv' ... doesn't seem to work. Any idea?

Thanks a lot!

 

[arildno]

Eeh, are talking about:
$$e^{2\ln(x)}$$
In that case, use the fact that ln(x) is the inverse function of e^x

 

[Wonderland]

Thank you, so e^2ln(x) = 2x?? I know the rule e^lna = a but here it is e^blna that makes me confused. Is the rule here e^(bln(a)) = ba?

 

[arildno]

Think about the following:

How can you rewrite, by rules for exponentiation for real numbers:
$$(a^{b})^{c}=??$$

 

[Wonderland]

a^bc of course

 

[arildno]

Of course!


So, if you have e^(2*ln(x)), can you utilize the above identity in a clever way to simplify your expression, say by starting with: e^(2*ln(x))=e^(ln(x)*2)

 

[Wonderland]

so here, it would be (e^2)(e^lnx) = x(e^2), you mean I should just see e^2 as a constant and taking it out of the integration?

 

[arildno]

so here, it would be (e^2)(e^lnx) = x(e^2), you mean I should just see e^2 as a constant and taking it out of the integration?

Eeh, no.
Think again.

 

[Wonderland]

Would it be (e^2)(e^lnx) = x^2? It doesn't seem right...

 

[Wonderland]

it would mean that e^blna = a^b

 

[arildno]

it would mean that e^blna = a^b

Yup!

 

[arildno]

Would it be (e^2)(e^lnx) = x^2? It doesn't seem right.
Your LHS is wrong, due to a faulty application of the rule.

e^(2*ln(x))=x^2

 

[Wonderland]

Phewww! Thanks!:-):-)

 

[arildno]

You're welcome!

 

[Wonderland]