Question

Which method transforms \[\int\frac{dx}{ax^2+bx+c}\] or \[\int\frac{dx}{\sqrt{ax^2+bx+c}}\] into an expression compatible with standard formulae?

Options

• Partial fractions only

• Writing \[x=a\sec\theta\] in every case

• Factorizing into \[(x-a)(x+a)\] only

• Completing the square

Answer 1

General quadratic expressions are reduced by completing the square. This rewrites \[ ax^2 + bx + c\] in a form involving \[ (x + h)^2 \pm k^2,\] after which a suitable substitution reduces the integral to a standard form.