Hyperbolic substitutions

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

For √(a²+x²), x=a sinh t avoids sign ambiguities because cosh t>0. For √(x²−a²), x=a cosh t works on x≥a; the negative branch needs a different setup.

Step-by-step example

1. Use x = sinh t.

   $$I=\int\frac{dx}{\sqrt{1+x^2}},\quad dx=\cosh t\,dt$$

2. The root is cosh t, which is positive.

   $$I=\int dt$$

3. Return using the inverse function.

   $$I=\operatorname{arsinh}x+C=\ln(x+\sqrt{1+x^2})+C$$