why cant we just use power rule for this
1/((x^2+4))^2
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.
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.)
To find the derivative of the function 1/((x^2+4))^2, the power rule alone cannot be applied directly in this case. The power rule states that the derivative of x^n with respect to x is nx^(n-1), where n is a constant exponent.
However, in our given function, the power of 2 is applied to the entire denominator (x^2+4) rather than just x. Therefore, we need to use a different approach to find the derivative.
To differentiate this function, we will use the chain rule. The chain rule allows us to differentiate composite functions by breaking them down into simpler components.
Let's go through the steps to find the derivative using the chain rule:
1. Rewrite the function as (x^2+4)^(-2).
2. Identify the outer function, which is the power of -2.
3. Differentiate the outer function: d/dx[(x^2+4)^(-2)] = -2(x^2+4)^(-2-1) * d/dx[(x^2+4)].
4. Differentiate the inner function: d/dx[(x^2+4)] = 2x.
5. Substitute the derivative of the inner function back into the equation:
-2(x^2+4)^(-2-1) * 2x.
6. Simplify further if required.
By following these steps and using the chain rule, we can find the derivative of the function 1/((x^2+4))^2.