Trigonometric substitution: three radical forms

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

For a>0 try x=a sin t, x=a tan t or x=a sec t respectively. Fix a branch and an interval before simplifying roots. This guide covers all three forms; the explicit engine method currently covers the first.

Step-by-step example

1. For √(a²−x²), choose −π/2<t<π/2.

   $$x=a\sin t,\quad dx=a\cos t\,dt$$

2. Transform the radical along with dx.

   $$\int\frac{dx}{\sqrt{a^2-x^2}}=\int dt$$

3. Return to the principal branch; |x|<a.

   $$I=\arcsin(x/a)+C$$