find all the indefinite integrals of a function in interval Well, this is the function: $$\frac{\left(5x+3\right)}{x^3+2x^2-3x}$$, and i need to find all the indefinite integrals in the interval $\left(1,\infty \right)$. 
So far i used integration by partial fractions to find the indefinite integrals  and i got this:$$2\ln \left(x-1\right)\:-\ln \left(x\right)\:-\ln \left(x+3\right)\:+\:C$$
From here i don't know what to do. can someone help me? tnx!
 A: You're done. If $F$ is an antiderivative for $f$ on an open interval $I$ (that is, $F'(x) = f(x)$ for all $x \in I$), then every antiderivative of $f$ on $I$ is of the form $F + c$, for $c \in \mathbb{R}$, and every function of this form is an antiderivative. 
Note, the fact that $I$ is an interval is crucial here. For example, consider the function $\frac{1}{x}$ on $\mathbb{R}^* = \mathbb{R}\setminus\{0\} = (-\infty, 0) \cup (0, \infty)$. As $F(x) = \ln|x|$ is an antiderivative for $f$ on $\mathbb{R}^*$, $\ln|x| + c$ is an antiderivative for $f$, but not every antiderivative of $f$ is of this form. For example, you can check that for any choice of $c_1, c_2 \in \mathbb{R}$,
$$G(x) = \begin{cases}
\ln|x| + c_1 & x > 0\\
\ln|x| + c_2 & x < 0
\end{cases}$$
is also an antiderivative for $f$. Unless $c_1 = c_2$, $G$ is not of the form $F + c$.
In general, if $U$ is an open set in $\mathbb{R}$, it can be written as union of $k$ disjoint open intervals where $k \in \mathbb{N}\cup\{\aleph_0\}$ (i.e. countably many intervals). To describe all the antiderivatives on $U$, you need to allow for the choice of $k$ arbitrary constants, one for each interval. To see this, note that if $F$ is an antiderivative, and $G$ is any other, then 
$$\frac{d}{dx}(G - F) = \frac{d}{dx}(G) - \frac{d}{dx}(G) = f - f = 0$$
so $F - G$ is locally constant (i.e. constant on every connected component). For an open subset $U$ of $\mathbb{R}$, its connected components are the aforementioned disjoint open intervals. Therefore, $G = F + C$ where $C$ is a locally constant function which is determined by the choice of constant on each of the disjoint open intervals.
