Finding the Length of a Helix Using U-Substitution

[BraedenP]

Homework Statement

I am trying to prove that the length of a helix can be represented by $2\pi=\sqrt{a^2+b^2}$

Homework Equations

The Attempt at a Solution

I have the following so far:

If the helix can be represented by $h(t)=a\cdot cos(t)+a\cdot sin(t)+b(t)$

Then the length is:
$$\int_{0}^{2\pi}\sqrt{(-a\cdot sin(t))^2+(a\cdot cos(t))^2+b^2}\;\: dt$$

My problem comes when integrating this. If I use the stuff in the root as u and do u-substitution, then du equals 0dt:

$$u=a^2sin^(t)+a^2cos^2(t)+b^2$$
$$du=(a^2sin(2t)-a^2sin(2t))dt=0dt$$

My logic fails me when figuring out how to continue from there. I need to somehow represent 1dt. How do I do this?

Help would be awesome!

 

[Dick]

If your du comes out to be zero, then u must be a constant. What constant is it? Use some trig to simplify your u.

 

[gb7nash]

You don't need u-substitution for this problem. Here's why:

Before you try taking the integral, inside the square root, you have a²sin²t + a²cos²t. Factor the a² out of both of them. What happens?

 

[BraedenP]

Yep, was definitely over-thinking it. Thanks guys :)