# 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.

$\displaystyle\int\sqrt{a^2-x^2}\,\textrm{dx}=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$

$\displaystyle\int\sqrt{x^2+a^2}\,\text{dx}=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C$

$\displaystyle\int\sqrt{x^2-a^2}\,\textrm{dx}=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right)+C$

• Try differentiating the right hand sides. – PM 2Ring Dec 5 '15 at 12:47
• For verification that the formulas are correct, try what PM 2Ring suggested, i.e., differentiate the right side. You can also just use integration by parts to get the closed forms of those integrals. Take the integrand as the first function and $x\mapsto 1$ as the second function during performing IBP. – learner Dec 5 '15 at 12:54

One may obtain these results by performing respectively the change of variable $x:=a\sin t$, $x:=a\sinh t$ and $x:=a\cosh t$.