Consider the function f(x)=2/x^2 −2/x^7 ,x>0. Let F(x) be the antiderivative of f(x) with F(1)=0. Then F(x)= ?

Question

Consider the function f(x)=2/x^2 −2/x^7 ,x>0.

Let F(x) be the antiderivative of f(x) with F(1)=0.

Then F(x)= ?

Answer 1

The given function:

f(x) = 2/x² - 2/x⁷

may be rewritten as follows:

f(x) = 2x^-2 - 2x^-7

Now, one finds the integral of this function applying the integration rules:

1. F(x) = ∫ f(x) dx
2. F(x) = ∫ (2x^-2 - 2x^-7) dx
3. F(x) = ∫ 2x^-2 dx - ∫ 2x^-7 dx
4. F(x) = 2 ∫ x^-2 dx - 2 ∫ x^-7 dx
5. F(x) = (2)(-1)x^-1 - (2)(-1/6)x^-6 + C
6. F(x) = -2x^-1 + (1/3)x^-6 + C

To find the constant C, one must use the information originally given in the problem. Which states, F(1) = 0; therefore substituting 1 for x and 0 for F(x) and solving for C:

1. -2x^-1 + (1/3)x^-6 + C = F(x)
2. -2(1)^-1 + (1/3)(1)^-6 + C = 0
3. -2 + 1/3 + C = 0
4. -5/3 + C = 0
5. C = 5/3

Using this value in the last equation one obtains:

1. F(x) = -2x^-1 + (1/3)x^-6 + C
2. F(x) = -2x^-1 + (1/3)x^-6 + 5/3

Final answer:

F(x) = -2x^-1 + (1/3)x^-6 + 5/3