\textbf{Primitive function} of function \[f(x)\] over some interval \[x\in[a, b]\] is a function \[F(x)\] whose derivative is the function \[f(x)\] on that interval.

,,\qquad \forall x\in[a, b],\quad F'(x) = f(x) 

\textbf{Antiderivative} (aka \textbf{indefinite integral}) of a function \[f(x)\] is a family of its primitive functions which differ by a constant \[C\in\mathbb{R}\]:

,,\qquad \int f(x)\mathrm{d}x = F(x) + C 

In a nutshell, integration is the opposite of differentiation:

,,\qquad \frac{\mathrm{d}\int f(x)\mathrm{d}x}{\mathrm{d}x} = F(x)

,,\qquad \mathrm{d}\int f(x)\mathrm{d}x = f(x)\mathrm{d}x

,,\qquad \int\mathrm{d}F(x) = F(x) + C

Solving integrals in general is pretty hard, but there are a lot of established ways to do it. As OP I'll post some of the standard approaches, but this thread is about any kind of integration so feel free to post integrals and theirs solutions.

Most basic methods are:

\begin{itemize}
\item Using the table of integrals
\item Using linearity property
\item Using substitution
\item Using partial integration
\item Reducing quadratic to its cannonical form
\item Partial decomposition
\end{itemize}