Find the interval on which the curve of y equals the integral from 0 to x of 2/(1+3t+t^2), dt is concave up.

how do i do this problem? i took notes on the lesson my course is telling me its from, but i have no idea how to go about it.

By the fundamental theorem of calculus, the first derivative of



is



Then the second derivative is



From here you can proceed with determining the concavity of by finding the possible inflection points (when or undefined). Since the denominator is positive for all , you know that is defined for all . So you're left with solving





So you have two intervals to consider, and . Check the sign of in both of these intervals; if on some interval, then is concave upward on that interval.