Question

Find the most general antiderivative of the function.
(x) = 3/5 - 8/x, x > 0

Answer 1

`F = (3)/(5) x - 8\cdot \ln |x| + C`

Explanation:

Let be
`f(x) = (3)/(5)-(8)/(x)` and
`F` is the antiderivative of
`f(x)` such that:

1)
`F = \int {\left((3)/(5)-(8)/(x) \right)} \, dx` Given.

2)
`F = (3)/(5) \int \, dx -8\int {(dx)/(x) }` (
`\int {[f(x)+g(x)]} \, dx = \int {f(x)} \, dx + \int {g(x)} \, dx`)

3)
`F = (3)/(5) x - 8\cdot \ln |x| + C`, where
`C` is the integration constant. (
`\int {k} \, dx = k\cdot x`;
`\int {(dx)/(x) } = \ln|x|`,
`\int {k\cdot f(x)} \, dx = k\int {f(x)} \, dx`) Result.