Find the derivative of the function.

Question

g'(x) = ?

Please show all work so I can get a better understanding for this type of problem

Answer 1

Just two points about this problem:

First, the derivative of the integral brings you back to where you started.

F(x) = integral of f(x) so F'(x) = f(x)

similarly g'(x) = (u^2-4)/(u^2+4) if g(x) = the integral of (u^2-4)/(u^2+4)

Secondly, (u^2-4)/(u^2+4) = 1 - 8/(u^2+4) Just do the long division or check it by putting 1 and -8/(u^2+4) over a common denominator:

1 = (u^2+4)/(u^2+4)

(u^2+4)/(u^2+4) - 8/(u^2+4) = (u^2+4-8)/(u^2+4) = (u^2-4)/(u^2+4)

Why this might be useful is there are integration formulas, such as number 76, if you happen to have a textbook "Calculus" 10th Ed., by Salas, Hille and Etgen. It gives the integral of du/(a^2 + u^2) as

(1/a)arctan(u/a)

Even if you get the wrong answer, it often gets you major partial credit points if you exhibit some knowledge of the calculus involved, such as the above.