<s>[INST] What are $\int\sqrt{a^2-x^2}\,\textrm{d}x, \int\sqrt{x^2+a^2}\,\textrm{d}x,\int\sqrt{x^2-a^2}\,\textrm{d}x$? Can someone confirm these equations below? I got it from my college textbook, unfortunately there are no proofs and more importantly I cannot seem to find any other sources that say have these equations.
$\displaystyle\int\sqrt{a^2-x^2}\,\textrm{dx}=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$
$\displaystyle\int\sqrt{x^2+a^2}\,\text{dx}=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C$
$\displaystyle\int\sqrt{x^2-a^2}\,\textrm{dx}=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right)+C$
 [/INST] It's OK to me.
If you just want to confirm it, you may just differentiate each function on the right side to see if you get the integrand on the left hand side.
One may obtain these results by performing respectively the change of variable
$x:=a\sin t$, $x:=a\sinh t$ and $x:=a\cosh t$.
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