4) On the integral from 1 to 2 ∫(4x^2+4)/x^2dx =?

just do a long division

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

or, 4 + 4/x^2 :-)

To solve the integral from 1 to 2 of (4x^2+4)/x^2 dx, you can use the properties of integrals to break it down into simpler integrals.

Step 1: Split the fraction into two terms:
∫(4x^2/x^2) dx + ∫(4/x^2) dx

Step 2: Simplify each term:
∫4 dx + ∫4/x^2 dx

Step 3: Integrate each term:
The integral of 4 dx is simply 4x.

The integral of 4/x^2 dx can be solved using the power rule for integration. The power rule states that the integral of x^n dx equals (x^(n+1))/(n+1), where n is any real number except -1. In this case, n = -2.

So, the integral of 4/x^2 dx is:
= 4 * (1/x^2)
= 4/x^2

Step 4: Evaluate the definite integral:
Evaluate both terms of the integral from 1 to 2.

For the first term, replace x with 2 and subtract the result when x is replaced with 1:
∫(4x^2/x^2) dx = [4x]^2_1 = 4(2) - 4(1) = 8 - 4 = 4

For the second term, replace x with 2 and subtract the result when x is replaced with 1:
∫(4/x^2) dx = [4/x]^2_1 = (4/2) - (4/1) = 2 - 4 = -2

Step 5: Combine the results:
Add the results from both terms:
4 + (-2) = 2

Therefore, the integral from 1 to 2 of (4x^2+4)/x^2 dx is equal to 2.