1. ## u-substitution with e?!

The question is:

and we have to use u-substitution to solve. I've tried using u = e^2 and du=e^2, however, this doesn't work out easily. Any suggestions?

2. ## Re: u-substitution with e?!

That's not do-able, check the spelling. And it needs a dx, yes?

3. ## Re: u-substitution with e?!

Ah, yes! Sorry about that. Here it is fixed.

4. ## Re: u-substitution with e?!

You surely do not mean " $u= e^2$" then? That is a constant. If you mean " $u= e^{2x}$" then $du= 2e^{2x}dx$. But you don't have " $e^{2x}$ in the numerator to use with the "dx".

Instead, write the integral as $\int_0^{ln(3)}\frac{e^x}{e^{2x}}dx- \int_0^{ln(3)}\frac{1}{e^{2x}}dx= \int_0^{ln(3)} e^{-x}dx- \int_0^{ln(3)} e^{-2x}dx$ and do the two integrals separately. If you need to, let u= -x in the first integral and let u= -2x in the second integral.

5. ## Re: u-substitution with e?!

Originally Posted by HallsofIvy
You surely do not mean " $u= e^2$" then? That is a constant. If you mean " $u= e^{2x}$" then $du= 2e^{2x}dx$. But you don't have " $e^{2x}$ in the numerator to use with the "dx".

Instead, write the integral as $\int_0^{ln(3)}\frac{e^x}{e^{2x}}dx- \int_0^{ln(3)}\frac{1}{e^{2x}}dx= \int_0^{ln(3)} e^{-x}dx- \int_0^{ln(3)} e^{-2x}dx$ and do the two integrals separately. If you need to, let u= -x in the first integral and let u= -2x in the second integral.
Okay, I'll try that. Thanks!

6. ## Re: u-substitution with e?!

Possibly a sub of u = e^x was intended. Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Full size

__________________________________________________ __________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!