Rationalizing radicals and fractional powers

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

For rational powers choose m as a common multiple of their denominators. Substitute dx and retain sign restrictions and real branches. A radical of an affine expression suggests t=√(ax+b).

Step-by-step example

1. For x>0 let t = √x.

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

2. Divide the resulting rational fraction.

   $$\frac{2t}{1+t}=2-\frac2{1+t}$$

3. Integrate and return to √x.

   $$I=2\sqrt x-2\ln(1+\sqrt x)+C$$