<s>[INST] Simple definite integral I do not figure out how to solve 
$$
L= \frac{1}{2}\int_1^4 \sqrt{(16t^2+t+4)}\;dt
$$
The key is probably in simplifying the polynomial equation, but I don't find a way that is really simplifying the integral. Can someone help?
Thank you.
 [/INST] Not that easy. I will ignore the 1/2 factor and the extrema, by now.
$$16t^2 + t + 4 = \left(4t + \frac{1}{8}\right)^2 + \frac{255}{64}$$
Substitute then $z = 4t + \frac{1}{8}$ and $dz = 4 dt$ you have now:
$$\frac{1}{4}\int\sqrt{z^2 + \frac{255}{64}} dz$$
Now use the substitution $z = \frac{1}{8}\sqrt{255}\tan(s)$ and $dz = \frac{1}{8}\sqrt{255}\sec^2(s) dx$ this will lead to (make the math):
$$\frac{\sqrt{255}}{{32}}\int \frac{1}{8}\sqrt{255}\sec^3(s) ds = \frac{255}{256}\int \sec^3(s) dx$$
Now use the secant reduction formula:
$$\int \sec^m(x) dx = \frac{\sin(x)\sec^{m-1}(x)}{m-1} + \frac{m-2}{m-1}\int \sec^{m-2}(x) dx$$
and in your case $m = 3$. Remember that the integral of $\sec(x)$ is $\log(\tan(x) + \sec(x))$ and then you get:
$$\frac{255}{512}\tan(s)\sec(s) + \frac{255}{512}\log(\tan(s) + \sec(s))$$
Substitute back now. You had:
$$s = \arctan\left(\frac{8u}{255}\right)$$
and 
$$u = \frac{1}{8} + 4y$$
obtaining:
$$\frac{1}{2}\left[\frac{1}{64}\sqrt{16t^2 + 4t + 4}(32t + 1) + \frac{255}{512}\log\left(\frac{8\sqrt{16t^2 + t + 4} + 32t + 1}{\sqrt{255}}\right)\right]_1^{4} = \text{do the math to obtain a number}$$
</s>