Challenging improper & definite integral

1. Jun 30, 2008

mathwizarddud

Evaluate

$$\int_{0}^{\infty} x\ (e^{3x}-1)^{-1/3}\ dx$$

and

$$\int_0^1\frac{x-1}{\ln\, x} \; dx$$

2. Jul 1, 2008

HallsofIvy

Staff Emeritus
Why? I presume you have some very good reason for wanting to do those integrals, perhaps because you need to learn how to do that sort of thing. In that case, it is far better for you to show what you have done so far so we can make suggestions on how to continue.

And if this is not homework, do you have any reason to believe it is possible to evaluate those other than numerically?

3. Jul 1, 2008

mathwizarddud

This is not homework of course. I know the answers already (I made one of them up), and I just want to see what would people say if they don't exactly know how to solve it (by hand) + if there's a more elegant way to solve these two so that I could learn something, too! LOL
:rofl:

4. Jul 1, 2008

rocomath

Yeah okay no one cares.

5. Jul 1, 2008

dirk_mec1

Hey I got a better idea why don't you show us how you solved them and then maybe we'll tell you if there's a shorter way home, m'kay?

6. Jul 3, 2008

Gib Z

Elegant solution for the second integral (at least in my opinion):

Identity: $$\int^b_a dx \int^t_{t_0} f(x,t) dt = \int^t_{t_0} dt \int^b_a f(x,t) dx$$

Consider $$\int^b_a dt \int^1_0 x^t dx$$.

Evaluating directly, we get $$\int^b_a \frac{1}{t+1} dt = \log_e \left( \frac{b+1}{a+1} \right)$$.

Applying the identity, we get $$\int^1_0 dx \int^b_a x^t dt$$, and evaluating the inner integral shows that is equal to $$\int^1_0 \frac{x^b -x^a}{\log_e x} dx$$.

Hence, $$\int^1_0 \frac{x^b-x^a}{\log_e x} dx = \log_e \left( \frac{b+1}{a+1} \right)$$

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EDIT: Whoops I forgot to evaluate the actual integral here. Letting b=1, a=0 gives us the original integral as equal to $\log_e 2$.

7. Jul 3, 2008

mathwizarddud

Yes, double integration is exactly what I had in mind.
Try the first one; it's a bit harder.

The answer is: $$(ln(3)+ \pi/3^{1.5})*\pi/3^{1.5}.$$