Question: The way I understand the definition of a combination of a set is that, without repetition,for example, the combination of the set $\{1,2\}$ would be $(1,2)$, and not $(1,1)$, $(1,2)$ and $(2,2)$ because I thought I read that each element has to be unique. Is this the case?

Also, 

What is the combination of a set that has only one element? like $\{2\}$ ?

the empty set? or is it $(2,2)$ which would to violate the unique property, if uniqueness actually is a constraint. I'm not sure if it is or not because the web doesn't cover the topic well in a 'non mathematician can understand it' way.

Answer: 
Let $w=$ some function of $u$ for which $dw = u\,du$.
You can take the solution as I've presented [here](https://math.stackexchange.com/questions/3057298/solving-used-real-based-methods-int-0x-fractk-lefttn-a-rightm-d):

\begin{equation}
\int\_0^x \frac{t^k}{\left(t^n + a\right)^m}\:dt= \frac{1}{n}a^{\frac{k + 1}{n} - m} \left[B\left(m - \frac{k + 1}{n}, \frac{k + 1}{n}\right) - B\left(m - \frac{k + 1}{n}, \frac{k + 1}{n}, \frac{1}{1 + ax^n} \right)\right]
\end{equation}

Here $a$ is $a^2$, $k = 1$, $m = \frac{3}{2}$, thus:

\begin{align}
\int\_0^x \frac{t}{\left(t^{2} + a^{2}\right)^{\frac{3}{2}}}\:dt &= \frac{1}{2}\left(a^2\right)^{\frac{1 + 1}{2} - \frac{3}{2}} \left[B\left(\frac{3}{2} - \frac{1 + 1}{2}, \frac{1 + 1}{2}\right) - B\left(\frac{3}{2} - \frac{1 + 1}{2}, \frac{1 + 1}{2}, \frac{1}{1 + a^2x^2} \right)\right] \\
&= \frac{1}{\left|a\right|}\left[B\left(\frac{1}{2}, 1\right) - B\left(\frac{1}{2},1, \frac{1}{1 + a^2x^2} \right) \right]
\end{align}

Using the relationship between the [Beta and the Gamma function](https://en.wikipedia.org/wiki/Beta_function#Relationship_between_gamma_function_and_beta_function) we find that:
\begin{equation}
 B\left(\frac{1}{2},1 \right) = \frac{\Gamma\left(\frac{1}{2} \right)\Gamma(1)}{\Gamma\left(1 + \frac{3}{2}\right)} = \frac{\Gamma\left(\frac{1}{2} \right)\Gamma(1)}{\frac{1}{2}\Gamma\left(\frac{1}{2}\right)} = 2\Gamma(1) = 2
\end{equation}

Thus, 

\begin{equation}
 \int\_0^x \frac{t}{\left(t^{2} + a^{2}\right)^{\frac{3}{2}}}\:dt = \frac{1}{\left|a\right|}\left[2 - B\left(\frac{1}{2},1, \frac{1}{1 + a^2x^2} \right) \right]
\end{equation}