`f(x) = 7x^(2/5) + 8x^(-4/5)` Find the most general antiderivative of the function.

Expert Answers
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The most general antiderivative F(x) of the function f(x) can be found using the following relation:

`int f(x)dx = F(x) + c`

`int (7x^(2/5) + 8x^(-4/5))dx = int (7x^(2/5))dx + int (8x^(-4/5))dx`

You need to use the following formulas:

`int x^n dx = (x^(n+1))/(n+1)`

`int x^(-n)dx = (x^(-n+1))/(-n+1)`

`int (7x^(2/5))dx = (x^(2/5+1))/(2/5+1) = (5/7)*x^(7/5) + c`

`int (8x^(-4/5))dx = (x^(-4/5+1))/(-4/5+1) = 5*x^(1/5) + c`

Gathering all the results yields:

`int (7x^(2/5) + 8x^(-4/5))dx = (5/7)*x^(7/5) + 5*x^(1/5) + c`

Hence, evaluating the most general antiderivative of the function yields `F(x) = (5/7)*x^(7/5) + 5*x^(1/5) + c.`