Euler substitutions for quadratic radicals

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Euler substitutions can convert rational functions of x and a quadratic radical into rational functions of a parameter. Variants use a>0, c>0 or a known real root. Each requires sign choices and nonzero denominators.

Step-by-step example

1. Choose t = √(x²+1)−x, so t>0.

   $$\sqrt{x^2+1}=x+t$$

2. Square and solve.

   $$x=\frac{1-t^2}{2t},\quad\sqrt{x^2+1}=\frac{1+t^2}{2t}$$

3. The differential transforms this integral into a logarithm.

   $$\int\frac{dx}{\sqrt{x^2+1}}=-\int\frac{dt}{t}=-\ln t+C$$