Question

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

Answer 1

Answer:

$F = \frac{3}{5} x - 8\cdot \ln |x| + C$

Step-by-step explanation:

Let be $f(x) = \frac{3}{5}-\frac{8}{x}$ and $F$ is the antiderivative of $f(x)$ such that:

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

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

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