Okay, how would you go about finding the area of a curve from 1 to 4, when y=2x+(2/(x^2))?? It's not like the problem I asked before because here, you cannot use substitution. I tried using 2x for u and x^2 for du but it won't simplify into a ln problem or anything that I can work with.

Also, how do you take the derivative/anti-derivative of a fraction like: (3x^2)/(2)?

I really appreciate the help!

To find the area under the curve defined by the equation y = 2x + (2 / (x^2)) from x = 1 to x = 4, you can use the definite integral.

Here's how you can approach it step by step:

1. Identify the function: In this case, the function is y = 2x + (2 / (x^2)).

2. Take the derivative of the function: To evaluate the derivative of a function like (3x^2) / 2, you can use the power rule. The power rule states that if you have a term of the form x^n, the derivative is nx^(n-1).

For example, if you want to find the derivative of (3x^2) / 2, you would first apply the power rule to x^2, which gives you 2x. Next, you move the constant term, 3/2, outside the derivative, so the final result is (3/2) * 2x, which simplifies to 3x.

3. Calculate the definite integral: To find the area under the curve, you need to evaluate the integral of the function from x = 1 to x = 4.

The definite integral is denoted as ∫[a,b] f(x) dx, where a and b represent the limits of integration and dx represents the differential with respect to x.

In this case, the definite integral to calculate the area under the curve would be: ∫[1,4] (2x + (2 / (x^2))) dx

4. Evaluate the integral: To find the antiderivative of the function (2x + (2 / (x^2))), you can split it into two separate integrals:

∫[1,4] 2x dx + ∫[1,4] (2 / (x^2)) dx

For the first integral, you can use the power rule again to find the antiderivative of 2x, which is x^2.

For the second integral, you can rewrite (2 / (x^2)) as 2x^(-2) and apply the power rule again to find the antiderivative, which is (-2 / x).

Once you have both antiderivatives, substitute the limits of integration:

[x^2] from 1 to 4 + [-2 / x] from 1 to 4

Evaluate each term, subtracting the value of the function at the lower limit (1) from the value at the upper limit (4):

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

Simplify the expression:

15 + (1/2)

Therefore, the area under the curve defined by y = 2x + (2 / (x^2)) from x = 1 to x = 4 is 15.5.

Remember to double-check the calculation for accuracy and make sure all steps have been followed correctly.