Differentiation

Inverse trig and hyperbolic functions in calculus

Differentiating $y=\arcsin x$ implicitly gives $\frac1{\sqrt{1-x^2}}$; similar results hold for $\arctan$, $\operatorname{arsinh}$, $\operatorname{arcosh}$. Read backwards, these give the integrals of $\frac1{\sqrt{a^2-x^2}}$, $\frac1{a^2+x^2}$ and $\frac1{\sqrt{x^2\pm a^2}}$, which algebra alone can't do.