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I know I must integrate $\int _0 ^1 uu_xu_{xx} dx$ by parts. I have $u = u$, $du = du$, $dv = u_xu_{xx}dx$, $v = \frac{1}{2} u^2_x$. With this, I end up with $\frac{1}{2}uu_x^2 |^1_0 - \frac{1}{2}\int u_x^2du$, but I have a note saying that the second term should be $u_x^3$, instead of $u_x^2$.

Also, was my process correct? I haven't integrated by parts in a very long time and it seems unfamiliar considering that $u=u(x,t)$.

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    The notation made it a mess. Let's say it's $\int_0^1 w w_x w_{xx} dx$. If $u = w$, then $du = w_x dx$. Do you see it now? – 2017-01-19
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    The mistake is $du=du$, when the correct would be $du=u_x\,dx$. Use the @OpenBall 's hint. – 2017-01-19
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    Yes, thank you so much for the clarification. The notation was really messing me up. Much appreciated – 2017-01-19

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You have done: $$ \int u u_xu_{xx}dx=\int u\cdot\,d\left(\frac{1}{2}u_x^2\right)= \frac{1}{2}uu_x^2-\int\frac{1}{2}u_x^2 du=\frac{1}{2}uu_x^2-\int\frac{1}{2}u_x^2 u_xdx $$ and the variable $t$ has no rule in this integration.