<s>[INST] 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.
 [/INST] 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.
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