The word "power rule" is usually associated with derivatives, but your subject title is "integrals"
so what do you want done with it
1/((x^2+4))^2
= (x^2 + 4)^-2
dy/dx = -2(x^2 + 4)^-3 (2x)
= -4x/(x^2 + 4)^3
if you want to integrate, it will be a bit harder.
why cant we just use power rule for this
1/((x^2+4))^2
2 answers
you can't use the power rule because that says
∫ u^n du = 1/(n+1) u^(n+1)
But if you let
u = x^2+4
du = 2x dx
So, you do not have
∫ u^-2 du
because you would need a 2x on top.
Instead you need to use a trig substitution. (Well, I guess you don't need to, but it makes the manipulations a whole lot easier.)
∫ u^n du = 1/(n+1) u^(n+1)
But if you let
u = x^2+4
du = 2x dx
So, you do not have
∫ u^-2 du
because you would need a 2x on top.
Instead you need to use a trig substitution. (Well, I guess you don't need to, but it makes the manipulations a whole lot easier.)