Question

why cant we just use power rule for this

1/((x^2+4))^2

Answer 1

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.

Answer 2

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 <b>need</b> to, but it makes the manipulations a whole lot easier.)