Two different integration questions are baffling me right now, and I have no idea how to approach them:
The first deals with finding the length of the curve $$ y = \int_{-2}^x sqrt(3t^4-1)dt$$
The second is (probably) a very simple integration problem that I can't get because my mind is now muddled with more complicated equations. It's part of a larger problem also involving length of a curve, but I can work it out after I figure this out:
$$ \int_t^1 sqrt(1+ (4/9)x^{(-2/3)})dx $$
Your help is much appreciated! I've been trying to solve them for the past hour and haven't gotten anywhere.
For your second problem, if $x$ is positive, we have $$\sqrt{1+\frac{4}{9}x^{-2/3}}=\frac{1}{3x^{1/3}}\sqrt{9x^{2/3}+4}.$$ To integrate, let $u=9x^{2/3}+4$. If we are integrating over an interval where $x$ is negative, $\frac{1}{3x^{1/3}}$ should be replaced by $-\frac{1}{3x^{1/3}}$.