How do you find the integral of sqrt[1+81x]? u-substitution and integration by parts does not seem to work.
On the contrary, u-sub does work. Try $\displaystyle u = 1+81x$
But for such a trivial integral, it's best to see it as $\displaystyle \frac{1}{81} \int \sqrt{1+81x}~d(1+81x)$ and integrate directly.
Are you sure you did not mean $\displaystyle \sqrt{1+81x^2}$ instead?