Trig substitution integral (I hope)

1. Jan 18, 2012

ArcanaNoir

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

I got to a place in a problem where I need to do a sticky integral, and I'm hoping I can use a trig substitution. If not, I will need to solve the main problem another way :(

$$\int_0^\infty \sqrt{1+(e^{-\theta })^2} \; \mathrm{d} \theta$$

2. Relevant equations

$$1+\tan ^2 \theta =\sec ^2 \theta$$

3. The attempt at a solution

Can I let $e^{- \theta } = \tan \phi$ ?
if so, does $\mathrm{d} \theta = \sec ^2 \phi \; \mathrm{d} \phi$ ?

And then, do I have
$$\int \sec ^3 \phi \; \mathrm{d} \phi$$ ?

2. Jan 18, 2012

Dick

Aside from the fact you can tell the integral is divergent just by looking at it, d(exp(-t))=(-exp(-t)*dt).

3. Jan 18, 2012

I like Serena

Any particular reason you want to do a trig substitution?

Myself, I'm not a fan of trig substitutions.
Usually my first step is take a part of the expression and call it "u".
Do a substitution and see what you are left with.
I see a trig substitution more as a last resort.

4. Jan 18, 2012

ArcanaNoir

Maybe I should start at the beginning.

I need to find the total length of the spiral $r(\theta )=e^{-\theta }$ for $\theta \in [0,\infty )$

There is a formula for arc length, but I don't necessarily have to use it if there is another way:

$$L=\int \sqrt{1+(r')^2} \; \mathrm{d} \theta$$

I'm concerned right away about the infinity.

Last edited: Jan 18, 2012
5. Jan 18, 2012

I like Serena

Hmm, that is not the right formula for the length of your curve.
I suspect you are mixing up the formula with a cartesian version.

6. Jan 18, 2012

ArcanaNoir

Alas. So, how about $$L=\int_0^\infty \sqrt{r^2+(\frac{dr}{d\theta} )^2} \; \mathrm{d} \theta$$

in which case I will be computing
$$\int_0^\infty \sqrt{ e^{-2 \theta } + (-e^{-\theta } )^2 } \; \mathrm{d} \theta$$ ?

7. Jan 18, 2012

I like Serena

That looks much better! ;)

8. Jan 18, 2012

ArcanaNoir

Okay, thanks. night-night!