AP Calculus AB and BC glossary

Completing the square

Completing the square rewrites an irreducible quadratic as a perfect square plus a constant. In integration it turns a denominator you cannot factor into the shape of the arctangent formula, making the antiderivative an inverse trigonometric function.

x^{2} + bx + c = \left(x + \frac{b}{2}\right)^{2} + \left(c - \frac{b^{2}}{4}\right)

For $\int \frac{dx}{x^{2}+2x+5}$ , completing the square gives $\left(x+1\right)^{2}+4$ , and the integral becomes $\frac{1}{2}\arctan\frac{x+1}{2} + C$ .

The mistake

Trying it on a quadratic that factors. If the denominator factors over the reals the problem wants partial fractions instead, and the two techniques are chosen by whether the discriminant is negative.

Appears in: Unit 6: Integration and Accumulation