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.

Question

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!

The indefinite integral of that function is
x^2 -2/x. Calculate the value of that function at x=4 and subtract the value at x=1. That will be the area above the x-axis.

Answer 1

To find the area under the curve y = 2x + 2/(x^2) from x = 1 to x = 4, you can use the definite integral. Let's walk through the steps:

1. Start by finding the antiderivative (indefinite integral) of the function with respect to x. For y = 2x + 2/(x^2), integrate each term separately:
∫(2x) dx = x^2 + C (C is the constant of integration)
∫(2/(x^2)) dx = -2/x + C

2. Next, evaluate the antiderivative at the upper and lower limits:
F(4) = (4)^2 - 2/(4) + C = 16 - 1/2 + C
F(1) = (1)^2 - 2/(1) + C = 1 - 2 + C

3. Subtract the evaluated antiderivatives for the upper and lower limits:
Area = F(4) - F(1) = (16 - 1/2 + C) - (1 - 2 + C)

Simplifying, the constant C cancels out, giving us the final result for the area under the curve.

Regarding the derivative and antiderivative of a fraction like (3x^2)/(2), the process is as follows:

To find the derivative of a fraction like (3x^2)/(2), you can use the power rule and the quotient rule together. The power rule states that if you have a term like x^n, the derivative is nx^(n-1). The quotient rule is used when you have a fraction of two functions.

Derivative:
1. Start by applying the power rule to the numerator: d/dx(3x^2) = 3(2)x^(2-1) = 6x.
2. Apply the power rule to the denominator: d/dx(2) = 0 (since 2 is a constant).
3. The derivative of the fraction (3x^2)/(2) is (6x)/(0) = 6x.

To find the antiderivative (indefinite integral) of a fraction like (3x^2)/(2), you can reverse the power rule:
1. Add 1 to the power of x: ∫(6x) dx = (6/2)x^2 + C = 3x^2 + C (where C is the constant of integration).

Note: In this case, since the denominator is a constant, the integral simplifies to just multiplying the numerator by (1/(2+1)) = 1/3.

I hope this explanation helps you! Let me know if you have any further questions.