Given the following derivatives, find the integrals 
Find the derivatives of $\ln(x+\sqrt{x^2+1})$ and $\arcsin(x)$, and use the result to find the integrals of the following functions:

*

*$$ \dfrac{1}{ \sqrt{ \pm x^2 \pm a^2 }} $$


*$$ \sqrt{\pm x^2 \pm a^2} $$
Except for the cases where both are minuses. $a$ is a positive constant.

So for the two derivatives, I just found the following derivatives $$[\ln (x+\sqrt{x^2 \pm a^2}) ]' = \dfrac{1}{\sqrt{x^2 \pm a^2}}$$
And also:
$$ [ b\arcsin(\dfrac{x}{a} + c)]' = \dfrac{b}{\sqrt{a^2-(x+ac)^2}}$$
These formules make the first part easy. We get $\int \dfrac{1}{ \sqrt{ x^2 + a^2 }} = \ln (x+\sqrt{x^2 + a^2})$, $\int \dfrac{1}{ \sqrt{ x^2 - a^2 }} = \ln (x+\sqrt{x^2 - a^2})$ and $\int \dfrac{1}{ \sqrt{ a^2 - x^2 }} = \arcsin(\dfrac{x}{a})$
However, I am not able to figure out what the easiest way is to get the second part of the question, using the knowledge we have. Can someone help out.

Hint. You may use an integration by parts for the second family:
$$
\begin{align}
\int \sqrt{\pm x^2 \pm a^2} \:{\rm{d}}x&=x\sqrt{\pm x^2 \pm a^2}-\int \frac{x \times\pm x }{\sqrt{\pm x^2 \pm a^2}}\:{\rm{d}}x\\
&=x\sqrt{\pm x^2 \pm a^2}-\int \frac{\pm x^2}{\sqrt{\pm x^2 \pm a^2}}\:{\rm{d}}x\\
&=x\sqrt{\pm x^2 \pm a^2}-\int \frac{\pm x^2 \pm a^2 -\pm a^2}{\sqrt{\pm x^2 \pm a^2}}\:{\rm{d}}x\\
&=x\sqrt{\pm x^2 \pm a^2}-\int \sqrt{\pm x^2 \pm a^2} \:{\rm{d}}x+\pm a^2\int \frac{1}{\sqrt{\pm x^2 \pm a^2}}\:{\rm{d}}x
\end{align}
$$ giving
$$
\int \sqrt{\pm x^2 \pm a^2} \:{\rm{d}}x=\frac x2\sqrt{\pm x^2 \pm a^2}+\frac{\pm a^2}{2}\int \frac{1}{\sqrt{\pm x^2 \pm a^2}}\:{\rm{d}}x
$$
then you conclude with the first family.
