Completing the square

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

A shift converts a quadratic into a standard form. The constant sign then determines whether an arctangent, a logarithm or a radical substitution is appropriate.

Step-by-step example

1. Complete the square.

   $$x^2+2x+5=(x+1)^2+4$$

2. Shift the variable.

   $$u=(x+1)/2,\quad dx=2\,du$$

3. Apply the arctangent rule.

   $$\int\frac{dx}{x^2+2x+5}=\frac12\arctan\frac{x+1}{2}+C$$