Hey I've been having trouble integrating:

$\displaystyle \int_0^1 \frac{\sin x}{x} dx$

Here's what I have tried so far:

$\displaystyle I = \int_0^1 \underbrace{\frac{1}{x}}_{\tfrac{d}{dx} \ln x} \sin x dx = \underbrace{\left. \ln x \cdot \sin x\right|_0^1}_0 - \int_0^1 \ln x \cdot \cos x dx $

$\displaystyle u = \ln x \rightarrow du = \frac{dx}{x} \ \ \ [Note: x = e^u]$

But here's where I encounter a problem...I can't break it down properly after I do that substitution ... I just keep getting 0 = 0 or something like that. Am I on the correct path though?